Introducing the global-warming exaggeration factor X

Guest essay by Christopher Monckton of Brenchley

Recently I provided – based on a characteristically interesting email from Roger Taguchi – a demonstration that IPCC has at least doubled true climate sensitivity. In this follow-up piece, will you please welcome the global-warming exaggeration factor X.

First, a breathless recap on my summary of Roger’s argument. Global temperature rose by 0.83 K from 1850-2016 (HadCRUT4), while CO₂ concentration rose from 280 to 400 ppmv (NCEI). Officially-predicted pre-feedback sensitivity ΔT₀ to this increase in CO₂ concentration is thus 0.312 [5.35 ln (400/280)] = 0.60 K. Even if CO₂ were the sole cause of all the warming, the post-feedback gain factor G would be 0.83/0.60 = 1.38. Then, since nearly all temperature feedbacks are short-acting, at doubled CO₂ concentration and after all feedbacks had acted, equilibrium sensitivity ΔT_eq would be only 0.312 x 5.35 ln (2) x 1.38 = 1.6 K. Yet the AR4, CMIP3 and CMIP5 central equilibrium-sensitivity predictions are of order 3.2 K.

Of course, not all feedbacks have acted yet, but, on the other side of the ledger, much of the warming since 1850 is attributable either to natural causes or to non-CO₂ manmade forcings. Netting off these two considerations, it is virtually certain that IPCC and the models are overestimating Man’s influence on climate by well over double.

In this follow-up posting, I shall put some numbers to this conclusion. As before, the official climate-sensitivity equation will illuminate the argument:

Fig. 1 The official climate-sensitivity equation. Pre-feedback sensitivity ΔT₀ = λ₀ ΔF. Post-feedback sensitivity ΔT is the product of ΔT₀ and the post-feedback gain factor G. By a suitable choice of the feedback sum, the equation can model transient or equilibrium climate sensitivity.

The official equation is simple. For instance, it has no term for ocean heat capacity and none for time-dependency. Yet, remarkably, it gives us the ballpark in which climate sensitivity will fall, and much can be learned from it, as a calibration step will demonstrate.

On the pre-feedback side, the values of the radiative forcing ΔF = 3.708 W m^–2 at doubled CO₂ and of the pre-feedback climate sensitivity parameter λ₀ = 0.312 K W^–1 m² are those that IPCC and nearly all general-circulation models use. Their product is pre-feedback climate sensitivity ΔT₀ = 1.16 K. Models take these values as near-constant in modern conditions.

As the bottom-left portion of the curve in Fig. 2 shows, this pre-feedback response to a radiative forcing occurs within a few years of the forcing.

Note that Roe (2009) uses 4 W m^–2 as the forcing at CO₂ doubling, rather than today’s IPCC estimate 3.7 W m^–2, so that his pre-feedback or reference climate sensitivity is 1.25 K.

It is at once evident that the bulk of the warming comes not from the direct CO₂ forcing but from consequential temperature feedbacks, particularly in the high-sensitivity case.

Therefore, the chief reason why climate sensitivity cannot be readily constrained is that the interval of models’ estimates for the feedback-sum Σ_ic_i is broad.

Fig. 2 Time-evolution (log time axis after 500 yr) of the probability distribution of future climate states (mean climate sensitivity and 95% confidence interval) generated from a simple climate model forced by a step-function radiative forcing ΔF = 4 W m^–2 at t = 0. Pre-feedback or reference climate sensitivity ΔT₀ = 1.25 K. At right is the equilibrium probability distribution ΔT_eq. Higher-sensitivity climates take longer to equilibrate. Based on Roe (2009, fig. 6).

As an input to the calibration step, Fig. 3, an enhanced detail from IPCC (2013, fig. 43a), shows (in gray annuli) the CMIP3 feedback-sum interval that was used for IPCC (2007):

Fig. 3 The feedback sum in the CMIP3 models was 1.93 [1.53, 2.35] W m^–2 K^–1

From the interval [1.53, 2.35] W m^–2 K^–1 of feedback sums Σ_ic_i, the feedback fraction f, which falls on the interval [–1, +1], may be derived by multiplying each value of the feedback-sum by λ₀ = 0.312 K W^–1 m², so that f falls on the interval [0.48, 0.73].

Since the post-feedback or system gain factor G is equal to (1 – f)^–1, G falls on [1.91, 3.74]. Equilibrium post-feedback climate sensitivity ΔT_eq, the product of ΔT₀ and G in the official climate-sensitivity equation shown in Fig. 1, falls on [2.2, 4.3] K, near-identical to the published CMIP3 equilibrium-sensitivity interval [2.1, 4.4] K (IPCC, 2013, p. 820, §9.7.3).

The calibration thus shows a remarkably close correspondence between the models’ output and the official equation’s output, demonstrating that, simple though it looks, it is capable of reproducing modelled climate sensitivity with some reliability.

Between the 2007 and 2013 Assessment Reports, IPCC made no change to the values of the CO₂ forcing or of the pre-feedback sensitivity parameter, leaving pre-feedback climate sensitivity unchanged at 0.312 x 3.708 = 1.16 K per CO₂ doubling.

Fig. 4 CMIP5 feedbacks scaled to climate sensitivity to demonstrate the extent to which the now-rejected French IPSL-CM5A-LR model was an outlier in the ensemble.

The CMIP5 models relied upon by IPCC (2013) substantially reduced the interval of the temperature-feedback sum Σ_ic_i from the 1.93 [1.53, 2.35] in CMIP3 to just 1.44 [1.00, 1.82] W m^–2 K^–1 in CMIP5, after exclusion of a single outlier model from France (Fig. 4).

Multiplying the new interval by λ₀, the interval of the feedback fraction f falls from 0.60 [0.48, 0.73] in CMIP3 to 0.45 [0.31, 0.57] in CMIP5, whereupon the interval of the post-feedback or system gain factor G falls from 2.50 [1.92, 3.71] to 1.81 [1.45, 2.33].

Equilibrium sensitivity should, therefore, have fallen from 3.2 [2.1, 4.4] K in the CMIP3 models for IPCC (2007) to just 2.1 [1.7, 2.7] K in the CMIP5 models, since the feedback-sum interval between the CMIP3 and CMIP5 models had fallen as shown in the enlargement from IPCC (2013, Fig. 9.43a) in Fig. 3.

However, the CMIP5 models held the central estimate of equilibrium climate sensitivity at 3.2 K per CO₂ doubling (IPCC, 2013, p. 83, box TFE.6), and the interval is [2.1, 4.7] K (ibid., ch. 9, p. 745).

Accordingly, the CMIP5 models on which IPCC (2013) relies, having failed to adjust sensitivity downward in line with the reduction in the feedback sum compared with CMIP3, are now – on this ground alone – overstating climate sensitivity by a factor 1.53 [1.25, 1.76].

With this necessary theoretical background, we can now return to the observed record. Last week’s calculations were based on the assumption that all of the 0.83 K global warming since 1850 was caused by CO₂.

However, IPCC’s various reports estimate that between 70% and 90% of all anthropogenic warming is attributable to CO₂.

Furthermore, an unknown fraction of the global warming since 1850 was natural. Legates et al. (2013, 2015), of which I was a co-author, determined from the datafile in Cook et al. (2013) that, of 11,944 papers on climate and related topics published in the learned journals in the 21 years 1992-2011, only 0.3% had actually stated that at least half the warming since 1950 was manmade.

To cover all possibilities, we shall model manmade fractions of warming at 10-100% of observed or predicted warming. But we shall not model Michael Mann’s assertion that there would have been significant global cooling but for manmade global warming, because IPCC’s conclusion, in each of its past three Assessment Reports, is that without manmade forcings there would have been little warming or cooling. See, for instance, Fig. 5.

Fig. 5 Land and ocean combined warming, 1900-2000 with (pink) and without (blue) manmade forcings, according to IPCC (2013, fig. SPM.6).

Those who want to push the envelope a la Mann may, of course, do so using the method described here. But, as Fig. 5 shows, they are not in the mainstream that they so much value.

The method of determining the X factor begins with the observation that pre-feedback warming only takes a few years to manifest itself fully following a forcing.

Therefore, little error arises from the assumption that transient and zero-feedback sensitivities are approximately equal.

In the words of Roger’s follow-up email:

“Since CO₂ levels have continually increased from year to year since 1850, it could be argued that we have never achieved a new steady state, so present temperature readings are lower than they would be at steady state (“equilibrium”).  This argument had some plausibility during the rapid temperature rise from about 1950 to 1998.

“But the 18-year hiatus in temperature rise is most probably explained by a relatively short time constant, so that present temperatures must be fairly close to steady-state temperatures, with maybe a time-lag of a couple of years.

“Why would people invoke ad-hoc time constants of centuries, and search for “missing heat”?  In order to save a failing theory, that doubling CO₂ results in an “equilibrium” climate sensitivity of 1 + 2 = 3 K after positive feedback owing to increased water vapor.”

The fraction of observed global warming that the increased partial pressure of CO₂ has generated since 1850 is by definition equal to the product of the anthropogenic fraction of all observed warming and the CO₂ fraction of the anthropogenic warming.

For instance, if 50% of all observed warming were anthropogenic and 70% of anthropogenic warming were CO₂-driven the fraction of observed warming since 1850 represented by CO₂ would be 50% of 70%, or 35%.

The next step is to determine the factor by which predicted pre-feedback CO₂-driven warming in response to the observed 120 ppmv growth in CO₂ concentration since 1850 exceeds the fraction of all observed warming since 1850 that is attributable to CO₂.

The product of this factor X₀ and the interval 1.53 [1.25, 1.76] for the excess feedback prediction factor X_f is the interval of exaggeration factors X by which the models on which IPCC relies have overstated CO₂-driven warming compared with observation since 1850.

The equation for the global-warming exaggeration factor X is thus as follows:

For instance, predicted CO₂-driven warming ΔT_0,pre since 1850 is 0.312 (5.35 ln 2) = 0.6 K; the fraction C of all warming since 1850 thought to be represented by CO₂ warming is, say, 0.35 (assuming, for illustration, that half of all warming since 1850 is anthropogenic and 70% of that warming was CO₂-driven, for 0.5 x 0.7 = 0.35); and the observed warming since 1850, assumed to be equal to the pre-feedback warming, is 0.83 K.

Then the pre-feedback exaggeration factor X₀ = 2.048.

Next, the published central CMIP5 estimate ΔT_eq,CMIP5 of equilibrium post-feedback warming is 3.2K; however, the central estimate of the system gain factor G_CMIP5, based on the published feedback-sums in Fig. 3, is 1.81, and the equilibrium pre-feedback warming in response to doubled CO₂ is 1.16 K.

Then the feedback exaggeration factor X_f = 1.53.

Accordingly X, the product of the pre-feedback and feedback exaggeration factors, equals 3.12, indicating that, assuming that warming to date is approximately equal to the pre-feedback response to radiative forcings, and assuming the chosen values for the fraction of all global warming since 1850 that was anthropogenic and for the fraction of all anthropogenic warming that was CO₂-driven, IPCC and the models are approximately tripling true climate sensitivity.

The equation for X can, of course, be adapted quite easily to allow for any degree of positive or negative feedback that is thought to have occurred over the period of study (in the present instance, 1850 to the present). But here, for simplicity, and because little error arises, it is assumed that warming to date is approximately equal to pre-feedback warming.

The beauty of this method is that a table of global-warming exaggeration factors X can be prepared as soon as the CO₂ and temperature values for a given month become available; and they can be computed for all anthropogenic-warming fractions and for all CO₂ fractions.

Results of just such a computation are shown in Table 1.

In Table 1, exaggeration factors X > 2.000, indicating that IPCC’s methods are at least doubling true climate sensitivity, are in red; exaggeration factors 0.000 < X < 2.000 are in orange; and exaggeration factors X < 0, indicating an understatement of true climate sensitivity, are in green.

There are remarkably few values X < 0. Very nearly all of the values in the table show exaggeration by the models, and the great majority of the exaggeration factors are substantial.

The heavy bias in the models towards exaggeration of climate sensitivity is thus at once visible and explicit in the table. It is indeed virtually certain that the models have exaggerated climate sensitivity by at least double.

Table 1 Global warming exaggeration factors, 1850-2016, based on the CMIP5 model ensemble. Red indicates exaggeration of climate sensitivity by double or more; yellow indicates exaggeration by less than double; green indicates understatement. For each block, the first line gives the factor by which pre-feedback climate sensitivity is exaggerated in models, and the following three lines encompass the interval of exaggeration factors based on CMIP5 models’ failure to take into account their own reduction in the feedback-sum interval when determining final climate sensitivity, with the central estimate in bold face.