Game over

*By Christopher Monckton of Brenchley*

Skeptics 1, Fanatics 0. That’s the final score.

The corrected mid-range estimate of Charney sensitivity, which is equilibrium sensitivity to doubled CO_{2} in the air, is less than half of the official mid-range estimates that have prevailed in the past four decades. Transient sensitivity of 1.25 K and Charney sensitivity of 1.45 K are nothing like enough to worry about.

This third article answers some objections raised as a result of the first two pieces. Before I give some definitions, equations and values to provide clarity, let me make it plain that my approach is to accept – for the sake of argument only – that everything in official climatology is true except where we have discovered errors. By this acceptance *solum ad argumentum,* we minimize the scope for futile objections that avoid the main point, and we focus the discussion on the grave errors we have found.

**Definitions**

All definitions except that of temperature feedback are mainstream. I am including them in the hope of forestalling comments to the effect that there is no such thing as the greenhouse effect, or that temperatures (whether entire or delta) cannot induce feedbacks. If you are already well versed in climatology, as most readers here are, skip this section except for the definition of feedback, where climatology is at odds with mainstream feedback theory.

*Greenhouse gases* possess at least three atoms in their molecules and are thus capable of possessing or, under appropriate conditions, acquiring a dipole moment that causes them to oscillate in one of their vibrational modes and thus to emit heat.

*Carbon dioxide *(CO_{2}), being symmetrical, does not possess a dipole moment, but acquires one in its bending vibrational mode on interacting with a near-infrared photon. To use Professor Essex’s excellent analogy, when a greenhouse gas meets a photon of the right wavelength it is turned on like a radiator, whereupon some warming must by definition occur.

*The non-condensing greenhouse gases* exclude water vapor.

*Water vapor, *the most significant greenhouse gas by quantity, is a condensing gas. All relevant changes in its atmospheric burden are treated as temperature feedbacks. Its atmospheric burden is thought to increase by 7% per Kelvin of warming in accordance with the Clausius-Clapeyron relation (Wentz 2007).

*Emission temperature* would obtain at the Earth’s surface if there were no non-condensing greenhouse gases or feedbacks present. Emission temperature is a function of insolation, albedo and emissivity (assumed to be unity), and of nothing else. As non-condensing greenhouse gases and feedbacks warm the atmosphere, the altitude at which the emission temperature obtains rises.

*Radiative forcing* (in W m^{–2}) is an exogenous perturbation in the net (down minus up) radiative flux density at the top of the atmosphere. Forcings become warmings via –

*The Planck sensitivity parameter *(in K W^{–1} m^{2}: Roe 2009), the quantity by which a radiative forcing is multiplied to yield the reference sensitivity. To a first approximation, it is the first derivative of the fundamental equation of radiative transfer with respect to the Earth’s emission temperature and emission flux density. Its value is thus dependent on insolation and albedo. The first derivative is the change in temperature per unit change in flux density, i.e., at today’s values 255.4 / (4 x 241.2) = 0.27 K W^{–1} m^{2}. However, owing to altitudinal variation, the modeled value today is 0.31 = 3.2^{–1} K W^{–1} m^{2} (IPCC 2007, p. 631 fn.).

*Temperature feedback* (in W m^{–2} K^{–1}), an additional forcing proportional to the temperature that induces it, in turn drives a feedback response (in K) that modifies the originating temperature. This definition of a feedback as a modification of a signal (not merely of a change in the input signal but also of the input signal itself) is standard in all applications of control theory except climatology, where it has been near-universally but falsely imagined that an input signal (emission temperature in the climate) does not induce a feedback, even where feedback processes are present and will modify even the tiniest change in that signal. It is this error that has misled official climatology into overestimating climate sensitivities.

Models do not implement feedback math explicitly. However, their outputs are routinely calibrated against past climate. Paper after paper incorrectly states that the entire 33 K difference between today’s surface temperature of 288 K and the emission temperature of 255 K that would prevail today in the absence of greenhouse gases or of feedbacks is driven by the directly-forced warming from the non-condensing greenhouse gases and the feedbacks induced by that warming.

For instance, Lacis (2010) says that three-quarters of the difference between emission temperature and today’s temperature is the feedback response to the non-condensing greenhouse gases: i.e, that the feedback fraction is 0.75, which, given the CMIP5 reference sensitivity of 1.1 K (Andrews 2012) would yield Charney sensitivity of 4.4 K. Sure enough, the CMIP5 models’ feedback fraction, at 0.67, is close to Lacis’ value, implying Charney sensitivity of 3.3 K. It will be proven that there is no justification whatever for mid-range estimates anything like this high. They arise solely because the models have been tuned over the decades to yield Charney sensitivities high enough to account for the entire 33 K.

*Reference sensitivity*is the temperature change in response to a radiative forcing before taking feedbacks into account.*Equilibrium sensitivity,*the warming expected to occur within a policy-relevant timeframe once the climate has resettled to equilibrium after perturbation by a radiative forcing (such as doubled CO_{2}concentration) and after all temperature feedbacks of sub-decadal duration have aced, may be somewhat larger than –*Transient climate sensitivity*, the warming expected to occur immediately in response to a forcing. The chief reason for the difference is the delay occasioned by the vast heat-sink that is the ocean.*Charney sensitivity,*named after Dr Jule Charney, is equilibrium sensitivity to doubled CO_{2}.

*Zero-dimensional-model equation *relates reference and equilibrium sensitivities or temperatures via the feedback fraction, which accounts for the entire difference between them. Control theory in all applications except climatology uses both forms of (1) and of its rearrangement, (2), but climatology has not hitherto appreciated that the right-hand form of each equation is permissible. For this reason, it has failed to accord sufficient – or in most instances any – weight to the feedback response that arises from the presence of emission temperature. As a result of this grave error, official climatology has greatly overestimated the feedback fraction and hence all transient and equilibrium climate sensitivities.

**Input variables**

Input variables** **are from official sources.** **Net industrial-era anthropogenic radiative forcing to 2011 was 2.29 W m^{–2} (IPCC 2013, table SPM.5); the Planck sensitivity parameter is 3.2^{–2} K W^{–1} m^{2} (IPCC 2007, p. 631 fn.); the radiative energy imbalance to 2010 was 0.59 W m^{–2} (Smith 2015); industrial-era warming to 2011 was 0.75 K (least-squares trend on the HadCRUT4 monthly global mean surface temperature anomalies, 1850-2011: Morice 2012); and the radiative forcing at CO_{2} doubling is 3.5 W m^{–2} (Andrews 2012); the Stefan-Boltzmann constant is 5.6704 x 10^{–8} W m^{–2} K^{–4} (Rybicki 1979); albedo without non-condensing greenhouse gases or feedbacks would be 0.418 (Lacis 2010); global mean surface temperature without greenhouse gases would be 252 K (*ibid.*); and today’s global mean surface temperature is 288.4 K (ISCCP 2016).

**Mid-range industrial-era Charney sensitivity**

Now for the simplest proof of small Charney sensitivity.** **Net industrial-era manmade forcing to 2011 was 2.29 W m^{–2}, implying industrial-era reference warming 2.29 / 3.2 = 0.72 K. The radiative imbalance to 2010 was 0.59 W m^{–2}. Warming has thus radiated 2.29 – 0.59 = 1.70 W m^{–2} (74.2%) to space. Equilibrium warming to 2011 may thus prove to have been 34.7% greater than the observed 0.75 K industrial-era warming to 2011. The feedback fraction for transient sensitivity is then *f* = 1 – 0.716 / 0.751 = 0.047, so that transient climate sensitivity is 1.09 / (1 – 0.047) = **1.15 K.** Industrial-era *f *for equilibrium sensitivity is 1 – 0.716 / (0.751 x 1.347) = 0.29, implying Charney sensitivity 1.09 / (1 – 0.29) = **1.55 K.**

That’s it. Charney sensitivity is less than half of the 3.3 K mid-range estimate in the CMIP3 and CMIP5 general-circulation models, distorted as they are by the long-standing misallocation of all 33 K of the difference between today’s temperature and emission temperature to greenhouse-gas forcings and consequent feedbacks.

**Mid-range pre-industrial Charney sensitivity**

To show how official climatology’s grave error arose, we shall study how it has been apportioning that 33 K difference between today’s temperature and emission temperature.

Lacis (2010) estimated albedo without greenhouse gases as 0.418, implying emission temperature [1364.625(1 – 0.418) / (4*σ*)]^{0.25} = 243.26 K (Stefan-Boltzmann equation, with unit emissivity). However, Lacis estimated the global mean surface temperature without non-condensing greenhouse gases as 252 K, implying a small feedback response to emission temperature, arising from melting equatorial ice and about 10% of the current atmospheric burden of water vapor. That 10% value can be obtained from the 7% per Kelvin increase in water vapor found in Wentz (2007): thus, 100 / 1.07^{33} = 10.7.

Global temperature in 1850 was 287.6 K. The 35.6 K difference between 287.6 and 252 K was given as 25% [8.9 K] directly-forced warming from the naturally-occurring, non-condensing greenhouse gases and 75% [26.7 K] feedback response to that greenhouse warming. However, if the feedback fraction *f* over Lacis’ 50-year study period were constant, for transient sensitivity *f* would be 1 – (243.26 + 8.9) / 287.6 = 0.123, and transient sensitivity itself would be 1.09 / (1 – 0.123) = **1.25 K.** If an energy imbalance in 1850 might eventually increase that year’s temperature by 10%, then *f* = 1 – (243.26 + 8.9) / (287.6 x 1.1) = 0.203. Charney sensitivity would then be 1.09 / (1 – 0.203) = **1.4 K. **

In Lacis, the 44.2 K difference between emission and 1850 temperatures comprises 8.7 K (3.6%) feedback response to the 243.3 K emission temperature and, since Lacis takes transient-sensitivity *f *= 0.75, directly-forced greenhouse warming of 8.9 K inducing 26.6 K (300%) feedback response. Thus, Lacis imagines the feedback responses to emission temperature and to direct greenhouse warming are 3.6% and 300% respectively of the underlying quantities, which is absurd. What is more, Lacis says that the feedback fraction 0.75 applies also to “current climate”, an explicit demonstration that climatology’s error leading to overstatements of equilibrium sensitivity in the models arose from its neglect of the large feedback response to emission temperature.

Our corrected method finds transient-sensitivity *f* a lot less that Lacis’ 0.75. It is just 0.123. Then the 44.2 K difference between 1850 temperature and emission temperature comprises 243.3 *f */ (1 – *f*) = 34.1 K feedback response to emission temperature; 8.9 K directly-forced greenhouse warming; and 8.9 *f */* *(1 – *f*) = 1.2 K feedback response to direct greenhouse warming. Thus, feedback responses to emission temperature and direct greenhouse warming are identical at *f */ (1 – *f*) = 14% of the underlying quantities.

In practice, ice-melt would steadily reduce the ice-covered surface area, reducing the surface-albedo feedback and hence the overall feedback fraction, though that effect might be largely canceled by increased water vapor and cloud feedback. The assumption of a uniform feedback fraction throughout the transition from emission temperature to 1850 temperature is, therefore, not unreasonable. Other apportionments might be made: but it would not be reasonable to make apportionments anywhere close to those of Lacis or of the CMIP models.

Note how well the industrial and pre-industrial sensitivities cohere, and how very much smaller they are than official climatology’s 0.67-075. The corrected industrial-era values, just **1.25 K** transient sensitivity and **1.55 K** equilibrium sensitivity, necessarily follow from the stated official definitions and values. In my submission, it is no longer legitimate for official climatology to maintain that the mid-range estimate of Charney sensitivity is anything like as high as the CMIP3/CMIP5 models’ 3.3 K.

**Certainty about uncertainties**

What of the uncertainties in our result? Some of the official input values on which we have relied are subject to quite wide error margins. However, because our mid-range estimate of Charney sensitivity is low, occurring at the left-hand end of the rectangular-hyperbolic curve of Charney sensitivities in response to various values of the feedback fraction, the interval of plausible sensitivities is nothing like as broad as the official interval, which I shall now demonstrate to be a hilarious fiction.

The Charney report of 1979, echoed by several IPCC *Assessment Reports, *gives a Charney-sensitivity interval **3.0 **[1.5, 4.5] K. The 2013 *Fifth Assessment Report *retains the bounds but no longer dares to state the mid-range estimate, for a reason that I shall now reveal.

By now it will be apparent to all that the chief uncertainty in deriving transient or equilibrium sensitivities is the value of the feedback fraction. I found it curious, therefore, that IPCC did not derive its mid-range estimate of Charney sensitivity from the mean of the bounds of the feedback fraction’s interval. The mismatch is quite striking (see below)

IPCC’s mid-range Charney sensitivity 3.0 K implies a feedback fraction 0.61, which is three times closer to the upper bound 0.74 than to the lower bound 0.23. If IPCC had derived its mid-range Charney sensitivity from a value of the feedback fraction midway between the bounds, its 3 K mid-range estimate would have fallen by an impressive 0.75 K to just 2.25 K:

How, then, did IPCC come to imagine that mid-range Charney sensitivity could be as high as 3 K? The Charney Report of 1979, the first official attempt to derive Charney sensitivity, provides a clue. On p. 9, Charney found that the interval was **2.4 **[1.6, 4.5] K, implying a feedback fraction close enough to the mean of its bounds. However, by p. 16 he had decided that his eponymous interval was “in the range 1.5-4.5 K, with the most probable value near 3 K”. Why did he go for 3 K? And why did IPCC and CMIP5 remain in that ballpark for four decades? Perhaps it was because, owing to their error, they could not otherwise account for the 33 K difference between emission temperature and present-day temperature.

Be that as it may, where (a) the feedback fraction is defined as 1 minus the ratio of reference to equilibrium temperature (Eq. (2)), where (b) the mid-range value of the feedback fraction is the mean of the bounds of its interval, and where (c) the mid-range estimate of equilibrium sensitivity is twice the lower-bound estimate, the upper bound of the feedback fraction must be unity. Then the upper bound of equilibrium sensitivity will fall precisely on the singularity in the rectangular-hyperbolic response curve, and will therefore be somewhere between plus and minus infinity (see above). This is definitive evidence that the supposed Charney-sensitivity interval **3.0 **[1.5, 4.5] K is nonsense, and that all attempts to ascribe a statistical confidence interval to it are likewise nonsense.

**Is our mid-range estimate of Charney sensitivity reasonable?**

Rud Istvan, in one of many interesting comments on the earlier articles, says Lewis & Curry (2014) found transient and equilibrium sensitivities to be 1.3 K and 1.65 K respectively, implying that Charney sensitivity is 1.25 times transient sensitivity, not 1.37 times as I calculated earlier. In that event, the feedback fraction is 1 – 0.716 / (0.751 x 1.25) = 0.237, implying Charney sensitivity 1.09 / (1 – 0.237) = **1.45 K, **similar to the **1.5 K** in Lewis 2015.

Rud offers the following interesting confirmatory method. In IPCC (2013), the mid-range estimates of the sub-decadal temperature feedback sum is 1.6 W m^{–2} K^{–1}, since the feedbacks other than the water-vapor feedback sum to zero. Multiplying the feedback sum by the Planck parameter gives a mid-range feedback fraction 0.5 (Table 1). Note in passing that, as discussed earlier, the upper-bound feedback fraction works out at the absurd value 1.0.

Rud goes on to point out that, as several papers show, the CMIP5 models produce about half the observed rainfall, implying that the modeled water-vapor feedback is double the true value. Therefore, he says, the true feedback fraction is half the CMIP5 models’ estimate. That means 0.25, giving a Charney sensitivity of 1.09 / (1 – 0.25) = **1.45 K.**

I shall let Rud Istvan have the last word:

“This is not coincidental. The ‘best’ Charney sensitivity, whether calculated using the energy budget, or observed v. modeled via Bode’s feedback fraction

f, is half of the ‘best estimate’ in IPCC (2007). I agree with Christopher Monckton of Brenchley. It’s game over.”

*Speaking of “game over”, it is not quite over yet for this friend of WUWT, who was recently featured in a top post. If you haven’t already, won’t you help reach the final goal?*

Superforest,Climate Change

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