Part III: How the feedback factor f was exaggerated
By Christopher Monckton of Brenchley
In this series (Part 1 and Part 2) I am exploring the cumulative errors, large and small, through which the climatological establishment has succeeded in greatly exaggerating climate sensitivity. Since the series concerns itself chiefly with equilibrium sensitivity, time-dependencies, including those arising from non-linear feedbacks, are irrelevant.
So far, it has been established that the models’ failure to determine the central estimate of equilibrium or final climate sensitivity ΔT from their central estimate of the unitless feedback factor f (see Part I of this series) combined with their erroneous official mixing of surface temperature and emission-altitude flux in the Stefan-Boltzmann equation to generate an excessive value for the climate-sensitivity parameter λ0 (see Part II) had led to a 40% exaggeration of the central estimates of the reference pre-feedback sensitivity ΔT0 and hence of final sensitivity ΔT in the CMIP5 ensemble of general-circulation climate models.
Part III will consider a further effect of the official exaggeration of λ0 on climate sensitivity –the overstatement of the temperature feedback factor f.
The official equation (1) of climate sensitivity as it now stands, which was well calibrated against the outputs of both the CMIP3 and CMIP5 model ensembles in Part I, is –
Fig. 1 Illumination of the official climate-sensitivity equation (1)
Fig. 1 illuminates the interrelation between the various terms in (1). We shall now determine equilibrium sensitivity stepwise, making corrections for the errors identified in Parts I and II along the way and, this time, also correcting the value of the feedback factor f.
The net incoming flux density F0, at the emission altitude about 5 km above ground level depends solely on the total solar irradiance S0 = 1361 W m–2 and on the mean albedo or reflectance α = 0.3, thus: F0 = S0(1 – α) / 4 = 238.175 W m–2. From the fundamental equation of radiative transfer, assuming emission-altitude emissivity ε0 = 1 and the Stefan-Boltzmann constant σ = 5.67 x 10–8 W m–2 K–4, emission temperature T0 = [F0 / (ε0 σ)]1/4 = 254.578 K.
Add the CO2 radiative forcing ΔF0 = 5.35 ln(2) = 3.708 W m–2 to obtain the pre-feedback or reference flux density Fμ = 241.883 W m–2, from which the Stefan-Boltzmann equation gives Tμ = 255.563 K, so that reference sensitivity ΔT0 = Tμ – T0 = 0.985 K.
With these preliminaries, we begin the consideration of temperature feedbacks, which are additional forcings ci, summing to c = Σi ci, expressed in Watts per square meter per Kelvin of the reference warming ΔT0 that triggered them. This time, we shall concentrate only on the central estimate of climate sensitivity. In the next article, we shall examine the upper and lower bounds, for the hitherto poorly-constrained breadth of the climate-sensitivity interval arises chiefly from variations in temperature feedbacks between models.
IPCC’s interval of climate sensitivities in AR5 is [1.5, 4.5] K, just as it was in the Charney report for the National Academy of Sciences in 1979. Where λ0 is the official reference-sensitivity parameter 3.2–1 K W–1 m2, the ratios G of these bounds to IPCC’s estimate ΔT0 = λ0ΔF0 = 1.159 K of reference sensitivity fall on [1.294, 3.883], implying [0.227, 0.742] as the bounds of the interval of feedback factors f = 1 – 1 / G. The central estimate of f is here taken simply as (0.227 + 0.742) / 2 = 0.485, implying a feedback sum c = f / λ0 = 1.550 W m–2 K–1.
At this stage we are not going to challenge IPCC’s implicit central estimate of the feedback sum. If we were to retain IPCC’s concept and estimate of λ0, and consequently its estimate of reference sensitivity ΔT0, then the central estimate of equilibrium sensitivity based on the feedback sum c = 1.550 W m–2 K–1 would be 2.2 K, as Fig. (1) shows.
Fig. 1 shows a stable, an unstable and a climate-unphysical region. The stable region, where the feedback factor is either negative or at most 0.1 (and preferably little more than 0.01), reflects the fact that process engineers designing electronic circuits designed to perform stably even where the reliability of componentry and the stability of ambient operating conditions cannot be guaranteed often use a rule-of-thumb maximum design value for feedbacks, since any value above the maximum may lead to unwanted instability.
Why is the operation of feedbacks in electronic circuits of interest when looking at the climate? The answer is that the mathematics of feedback amplification was originally developed for electronic circuits, typically amplifiers, and that the two papers that between them established the present mathematical approach to feedbacks in the climate – Hansen (1984) and Schlesinger (1985) – refer back specifically to the treatment of feedbacks in electronic circuits as the origin of and justification for the method they proposed.
Fig. 1 The rectangular-hyperbolic curve of equilibrium climate sensitivity ΔT in response to the feedback factor f = λ0Σici, based on the official method of determining climate sensitivity, showing that implicit official final sensitivity in response to the central estimate f = 0.485 is 2.2 K.
Now, the mere fact that process engineers often try to impose an upper bound on feedback where it might lead to instability does not prove that climate feedbacks in the region shown in Fig. 1 as unstable are impossible. However, it suggests that they are unlikely; and, in the next article, we shall demonstrate that, in the climate, feedbacks do not occur in that region, and that they only appear to do so owing to a substantial error in climate feedback analysis.
For now, we shall take IPCC’s implicit central estimate of the feedback sum c = 1.550 W m–2 K–1 and use it as the basis for determining the central estimate of climate sensitivity, but without using the defective official quantity λ0.
Instead, we shall redetermine the unitless feedback factor f as the product of c and the first derivative of the Stefan-Boltzmann equation at the emission altitude after taking into account the pre-feedback increase in radiative flux density at that altitude, thus:
From this value, the final gain factor G = 1 / (1 – f ) = 1.693. The product of G and ΔF0 gives the final flux change ΔF, so that the final flux density F = F0 + ΔF = 244.454 W m–2, whereupon the final temperature T is 256.239 K, and the final sensitivity ΔT = T – T0 < 1.7 K.
Charney (1979) gave the central estimate of ΔT as 3.0 K. The CMIP5 models’ value is 3.2 K, which is a 92.5% exaggeration compared with the value 1.661 K found here. As we shall see later in the series, even this corrected central estimate is substantially too high.
Table 1 summarizes the calculations in this article.
|Determination of the central estimate of final climate sensitivity|
|2 x CO2 forcing ΔF0||5.35 ln (2)||3.708||W m–2|
|Emission flux density F0||S0 (1 – α) / 4||238.175||W m–2|
|Amplified flux density Fμ||F0 + ΔF0||241.883||W m–2|
|Amplified temperature Tμ||(Fμ / σ)1/4||255.563||K|
|Emission temperature T0||(F0 / σ)1/4||254.578||K|
|Reference sensitivity ΔT0||Tμ – T0||0.985||K|
|Official feedback factor foff||(0.227 + 0.742) / 2||0.485||Unitless|
|Implicit feedback sum c||foff / λ0 | λ0 = 3.2–1||1.550||W m–2 K–1|
|Corrected feedback factor f||c Tμ / (4Fμ)||0.409||Unitless|
|Final gain factor G||(1 – f )–1||1.693||Unitless|
|Final flux change ΔF||G ΔF0||6.279||W m–2|
|Final flux density F||F0 + ΔF||244.454||W m–2|
|Final temperature T||(F / σ)1/4||256.239||K|
|Final sensitivity ΔT||T – T0||1.661||K|
Ø Next: How the breadth of the climate-sensitivity interval was exaggerated.
Charney J (1979) Carbon Dioxide and Climate: A Scientific Assessment: Report of an Ad-Hoc Study Group on Carbon Dioxide and Climate, Climate Research Board, Assembly of Mathematical and Physical Sciences, National Research Council, Nat. Acad. Sci., Washington DC, July, pp. 22
Hansen J, Lacis A, Rind D, Russell G, Stone P, Fung I, Ruedy R, Lerner J (1984) Climate sensitivity: analysis of feedback mechanisms. Meteorol. Monographs 29:130–163
IPCC (1990-2013) Assessment Reports AR1-5 are available from www.ipcc.ch
Schlesinger ME (1985) Quantitative analysis of feedbacks in climate models simulations of CO2-induced warming. In: Physically-Based Modelling and Simulation of Climate and Climatic Change – Part II (Schlesinger ME, ed.), Kluwer Acad. Pubrs. Dordrecht, Netherlands, 1988, 653-735.