Part II: How the central estimate of pre-feedback warming was exaggerated
By Christopher Monckton of Brenchley
In this series 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.
In Part I, I described a small error by which the climate establishment determines the official central estimate of equilibrium climate sensitivity as the inter-model mean equilibrium sensitivity rather than determining that central estimate directly from the inter-model mean value of the temperature feedback factor f. For it is the interval of values for f that dictates the interval of final or equilibrium climate sensitivity and accounts for its hitherto poorly-constrained breadth [1.5, 4.5] K. Any credible probability-density function for final sensitivity must, therefore, center on the inter-model mean value of f, and not on the inter-model mean value of ΔT, skewed as it is by the rectangular-hyperbolic (and hence non-linear) form of the official system gain equation G = (1 – f)–1.
I showed that the effect of that first error was to overstate the key central estimates of final sensitivity by between 12.5% and 34%.
Part II, which will necessarily be lengthy and full of equations, will examine another apparently small but actually significant error that leads to an exaggeration of reference or pre-feedback climate sensitivity ΔT0 and hence of final sensitivity ΔT.
For convenience, the official equation (1) of climate sensitivity as it now stands is here repeated. There is much wrong with this equation, but, like it or not, it is what the climate establishment uses. In Part I, it was calibrated closely and successfully against the outputs of both the CMIP3 and CMIP5 model ensembles.
Fig. 1 illuminates the interrelation between the various terms in (1). In the current understanding, the reference or pre-feedback sensitivity ΔT0 is simply the product of the official value of radiative forcing ΔF0 = 3.708 W m–2 and the official value of the reference sensitivity parameter λ0 = 3.2–1 K W–1 m2, so that ΔT0 = 1.159 K (see e.g. AR4, p. 631 fn.).
However, as George White, an electronics engineer, has pointed out (pers. comm., 2016), in using a fixed value for the crucial reference sensitivity parameter λ0 the climate establishment are erroneously treating the fourth-power Stefan-Boltzmann equation as though it were linear, when of course it is exponential.
This mistreatment in itself leads to a small exaggeration, as I shall now show, but it is indicative of a deeper and more influential error. For George White’s query has led me to re-examine how, in official climatology, λ0 came to have the value at or near 0.312 K W–1 m2 that all current models use.
Fig. 1 Illumination of the official climate-sensitivity equation (1)
The fundamental equation (2) of radiative transfer relates flux density Fn in Watts per square meter to the corresponding temperature Tn in Kelvin at some surface n of a planetary body (and usually at the emission surface n = 0):
where the Stefan-Boltzmann constant σ is equal to 5.6704 x 10–8 W m–2 K–4, and the emissivity εn of the relevant surface n is, by Kirchhoff’s radiation law, equal to its absorptivity. At the Earth’s reference or emission surface n = 0, a mean 5.3 km above ground level, emissivity ε0, particularly with respect to the near-infrared long-wave radiation with which we are concerned, is vanishingly different from unity.
The Earth’s mean emission flux density F0 is given by (3),
where S0 = 1361 W m–2 is total solar irradiance (SORCE/TIM, 2016); α = 0.3 is the Earth’s mean albedo, and 4 is the ratio of the surface area of the rotating near-spherical Earth to that of the disk that the planet presents to incoming solar radiation. Rearranging (2) as (4) and setting n = 0 gives the Earth’s mean emission temperature T0:
A similar calculation may be performed at the Earth’s hard-deck surface S. We know that global mean surface temperature TS is 288 K, and measured emissivity εS ≈ 0.96. Accordingly, (3) gives FS as 374.503 W m–2. This value is often given as 390 W m–2, for εS is frequently taken as unity, since little error arises from that assumption.
The first derivative λ0 of the Stefan-Boltzmann equation relating the emission temperature T0 to emission flux density F0 before any radiative perturbation is given by (5):
The surface equivalent λS = TS / (4FS) = 0.192 K W–1 m2 (or 0.185 if εS is taken as unity).
The official radiative forcing in response to a doubling of atmospheric CO2 concentration is given by the approximately logarithmic relation (6) (Myhre et al., 1998; AR3, ch. 6.1). We shall see later in this series that this value is an exaggeration, but let us use it for now.
Then the direct or reference warming in response to a CO2 doubling is given by
A similar result may be obtained thus: where Fμ = F0 + ΔF0 = 238.175 + 3.708 = 241.883 W m–2, using (2) gives Tμ:
a little less than the result in (7), the small difference being caused by the fact that λ0 cannot have a fixed value, because, as George White rightly points out, it is the first derivative of a fourth-power relation and hence represents the slope of the curve of the Stefan-Boltzmann equation at some particular value for radiative flux and corresponding value for temperature.
Thus, the value of λ0, and hence that of climate sensitivity, must decline by little and little as the temperature increases, as the slightly non-linear curve in Fig. 2 shows.
Fig. 2 The first derivative λ0 = T0 / (4F0) of the Stefan-Boltzmann equation, which is the slope of a line tangent to the red curve above, declines by little and little as T0, F0 increase.
The value of λ0 may also be deduced from eq. (3) [here (10)] of Hansen (1984), who says [with notation altered to conform to the present work]:
“… for changes of solar irradiance,
“Thus, if S0 increases by a small percentage δ, T0 increases by δ/4. For example, a 2% change in solar irradiance would change T0 by about 0.5%, or 1.2-1.3 K.”
Hansen’s 1984 paper equated the radiative forcing ΔF0 from a doubled CO2 concentration with a 2% increase ΔF0 = 4.764 W m–2 in emission flux density, which is where the value 1.2-1.3 K for ΔT0 = ΔF0λ0 seems first to have arisen. However, if today’s substantially smaller official value ΔF0 = 3.708 W m–2 (Myhre et al., 1998; AR3, ch. 6.1) is substituted, then by (10), which is Hansen’s equation, ΔT0 becomes 0.991 K, near-identical to the result in (7) here, providing further confirmation that the reference or pre-feedback temperature response to a CO2 doubling should less than 1 K.
The Charney Report of 1979 assumed that the entire sensitivity calculation should be done with surface values FS, TS, so that, for the 283 K mean surface temperature assumed therein, the corresponding surface radiative flux obtained via (2) is 363.739 W m–2, whereupon λS was found equal to a mere 0.195 K W–1 m2, near-identical to the surface value λS = 0.192 K determined from (5).
Likewise, Möller (1963), presenting the first of three energy-balance models, assumed today’s global mean surface temperature 288 K, determined from (2) the corresponding surface flux 390 W m–2, and accordingly found λS = 288 / (4 x 390) = 0.185 K W–1 m2, under the assumption that surface emissivity εS was equal to unity.
Notwithstanding all these indications that λ0 is below, and perhaps well below, 0.312 K W–1 m2 and is in any event not a constant, IPCC assumes this “uniform” value, as the following footnote from AR4, p.631, demonstrates [with notation and units adjusted to conform to the present series]:
“Under these simplifying assumptions the amplification of the global warming from a feedback parameter c (in W m–2 K–1) with no other feedbacks operating is 1 / (1 – c λ0), where λ0 is the ‘uniform temperature’ radiative cooling response (of value approximately 3.2–1 K W–1 m2; Bony et al., 2006). If n independent feedbacks operate, c is replaced by (c1 + c2 +… + cn).”
How did this influential error arise? James Hansen, in his 1984 paper, had suggested that a CO2 doubling would raise global mean surface temperature by 1.2-1.3 K rather than just 1 K in the absence of feedbacks. The following year, Michael Schlesinger described the erroneous methodology that permitted Hansen’s value for ΔT0 to be preserved even as the official value for ΔF0 fell from Hansen’s 4.8 W m–2 per CO2 doubling to today’s official (but still much overstated) 3.7 W m–2.
In 1985, Schlesinger stated that the planetary radiative-energy budget was given by (11):
where N0 is the net radiation at the top of the atmosphere, F0 is the downward flux density at the emission altitude net of albedo as determined in (3), and R0 is the long-wave upward flux density at that altitude. Energy balance requires that N0 = 0, from which (3, 4) follow.
Then Schlesinger decided to express N0 in terms of the surface temperature TS rather than the emission temperature T0 by using surface temperature TS as the numerator and yet by using emission flux F0 in the denominator of the first derivative of the fundamental equation (2) of radiative transfer.
In short, he was applying the Stefan Boltzmann equation by straddling uncomfortably across two distinct surfaces in a manner never intended either by Jozef Stefan (the only Slovene after whom an equation has been named) or his distinguished Austrian pupil Ludwig Boltzmann, who, 15 years later, before committing suicide in despair at his own failure to convince the world of the existence of atoms, had provided a firm theoretical demonstration of Stefan’s empirical result by reference to Planck’s blackbody law.
Since the Stefan-Boltzmann equation directly relates radiative flux and temperature at a single surface, the official abandonment of this restriction – which has not been explained anywhere, as far as I can discover – is, to say the least, a questionable novelty.
For we have seen that the Earth’s hard-deck emissivity εS is about 0.96, and that its emission-surface emissivity ε0, particularly with respect to long-wave radiation, is unity. Schlesinger, however, says:
“N0 can be expressed in terms of the surface temperature TS, rather than [emission temperature] T0 by introducing an effective planetary emissivity εp, in (12):
so that, in (13),
This official approach embodies a serious error arising from a misunderstanding not only of (2), which relates temperature and flux at the same surface and not at two distinct surfaces, but also of the fundamental architecture of the climate.
Any change in net flux density F0 at the mean emission altitude (approximately 5.3 km above ground level) will, via (2), cause a corresponding change in emission temperature T0 at that altitude. Then, by way of the temperature lapse rate, which is at present at a near-uniform 6.5 K km–1 just about everywhere (Fig. 3), that change in T0 becomes an identical change TS in surface temperature.
Fig. 3 Altitudinal temperature profiles for stations from 71°N to 90°S at 30 April 2011, showing little latitudinal variation in the lapse-rate of temperature with altitude. Source: Colin Davidson, pers. comm., August 2016.
But what if albedo or cloud cover or water vapor, and hence the lapse rate itself, were to change as a result of warming? Any such change would not affect the reference temperature change ΔT0: instead, it would be a temperature feedback affecting final climate sensitivity ΔT.
The official sensitivity equation thus already allows for the possibility that the lapse-rate may change. There is accordingly no excuse for tampering with the first derivative of the Stefan-Boltzmann equation (2) by using temperature at one altitude and flux at quite another and conjuring into infelicitous existence an “effective emissivity” quite unrelated to true emissivity and serving no purpose except unjustifiably to exaggerate λ0 and hence climate sensitivity.
One might just as plausibly – and just as erroneously – choose to relate emission temperature with surface flux, in which event λ0 would fall to 254.6 / [4(390.1)] = 0.163 K W–1 m2, little more than half of the models’ current and vastly-overstated value.
This value 0.163 K W–1 m2 was in fact obtained by Newell & Dopplick (1979), by an approach that indeed combined elements of surface flux FS and emission temperature T0.
The same year the Charney Report, on the basis of hard-deck surface values TS and FS for temperature and corresponding radiative flux density respectively, found λS to be 0.192 K W–1 m2.
IPCC, followed by (or following) the overwhelming majority of the models, takes 3.2–1, or 0.3125, as the value of λ0. This choice thus embodies two errors one of modest effect and one of large, in the official determination of λ0. The error of modest effect is to treat λ0 as though it were constant; the error of large effect is to misapply the fundamental equation of radiative transfer by straddling two distinct surfaces in using it to determine λ0. As an expert reviewer for AR5, I asked IPCC to provide an explanation showing how λ0 is officially derived. IPCC curtly rejected my recommendation. Perhaps some of its supporters might assist us here.
In combination, the errors identified in Parts I and II of this series have led to a significant exaggeration of the reference sensitivity ΔT0, and commensurately of the final sensitivity ΔT, even before the effect of the errors on temperature feedbacks is taken into account. The official value ΔT0 = 1.159 K determined by taking the product of IPCC’s value 0.3125 K W m–1 for λ0 and its value 3.708 W m–2 for ΔF0 is about 17.5% above the ΔT0 = 0.985 K determined in (9).
Part I of this series established that the CMIP5 models had given the central estimate of final climate sensitivity ΔT as 3.2 K when determination of the central estimate of final sensitivity from the inter-model mean central estimate of the feedback factor f would mandate only 2.7 K. The CMIP 5 models had thus already overestimated the central estimate of equilibrium climate sensitivity ΔT by about 18.5%.
The overstatement of the CMIP5 central estimate of climate sensitivity resulting from the combined errors identified in parts I and II of this series is accordingly of order 40%.
This finding that the current official central estimate climate sensitivity is about 40% too large does not yet take account of the effect of the official overstatement of λ0 on the magnitude of that temperature feedback factor f. We shall consider that question in Part III.
For now, the central estimate of equilibrium climate sensitivity should be 2.3 K rather than CMIP5’s 3.2 K. Though each of the errors we are finding is smallish, their combined influence is already large, and will become larger as the compounding influence of further errors comes to be taken into account as the series unfolds.
Table 1 shows various values of λ0, compared with the reference value 0.264 K W–1 m2 obtained from (8).
|Table 1: Some values of the reference climate-sensitivity parameter λ0|
|Source||Method||Value of λ0||x 3.7 = ΔT0||Ratio|
|Newell & Dopplick (1979)||T0 / (4FS)||0.163 K W–1 m2||0.604 K||0.613|
|Möller (1963)||TS / (4FS)||0.185 K W–1 m2||0.686 K||0.696|
|Callendar (1938)||TS / (4FS)||0.195 K W–1 m2||0.723 K||0.734|
|From (8) here||T0 / (4F0)||0.264 K W–1 m2||0.985 K||1.000|
|Hansen (1984)||T0 / (4F0)||0.267 K W–1 m2||0.990 K||1.005|
|From (7) here||T0 / (4F0)||0.267 K W–1 m2||0.991 K||1.006|
|Schlesinger (1985)||TS / (4F0)||0.302 K W–1 m2||1.121 K||1.138|
|IPCC (AR4, p. 631 fn.)||3.2–1||0.312 K W–1 m2||1.159 K||1.177|
Nearly all models adopt values of λ0 that are close to or identical with IPCC’s value, which appears to have been adopted for no better reason that it is the reciprocal of 3.2, and is thus somewhat greater even than the exaggerated value obtained by Schlesinger (1985) and much copied thereafter.
In the next instalment, we shall consider the effect of the official exaggeration of λ0 on the determination of temperature feedbacks, and we shall recommend a simple method of improving the reliability of climate sensitivity calculations by doing away with λ0 altogether.
I end by asking three questions of the Watts Up With That community.
1. Is there any legitimate scientific justification for Schlesinger’s “effective emissivity” and for the consequent determination of λ0 as the ratio of surface temperature to four times emission flux density?
2. One or two commenters have suggested that the Stefan-Boltzmann calculation should be performed entirely at the hard-deck surface when determining climate sensitivity and not at the emission surface a mean 5.3 km above us. Professor Lindzen, who knows more about the atmosphere than anyone I have met, takes the view I have taken here: that the calculation should be performed at the emission surface and the temperature change translated straight to the hard-deck surface via the lapse-rate, so that (before any lapse-rate feedback, at any rate) ΔTS ≈ ΔT0. This implies λ0 = 0.264 K W–1 m2, the value taken as normative in Table 1.
3. Does anyone here want to maintain that errors such as these are not represented in the models because they operate in a manner entirely different from what is suggested by the official climate-sensitivity equation (1)? If so, I shall be happy to conclude the series in due course with an additional article summarizing the considerable evidence that the models have been constructed precisely to embody and to perpetuate each of the errors demonstrated here, though it will not be suggested that the creators or operators of the models have any idea that what they are doing is as erroneous as it will prove to be.
Ø Next: How temperature feedbacks came to be exaggerated in official climatology.
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SORCE/TIM latest quarterly plot of total solar irradiance, 4 June 2016 to 26 August 2016. http://lasp.colorado.edu/data/sorce/total_solar_irradiance_plots/images/tim_level3_tsi_24hour_3month_640x480.png, accessed 3 September 2016
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