Guest essay By Christopher Monckton of Brenchley
This series discusses climatology’s recently-discovered grave error in having failed to take due account of the large feedback response to emission temperature. Correct the error and global warming will be small, slow, harmless and net-beneficial. The series continues to attract widespread attention, not only here but elsewhere. The ripples are spreading.
My reply to Roy Spencer’s piece on our discovery at drroyspencer.com has attracted 1400 hits, and the three previous pieces here have attracted 1000+, 350+ and 750+ respectively. Elsewhere, a notoriously irascible skeptical blogger, asked by one of his followers whether he would lead a thread on our result, replied that he did not deign to discuss anything so simple. Simple it is. How could it have been thought the feedback processes in the climate would not respond to the large pre-existing emission temperature to the same degree as they respond to the small enhancement of that temperature caused by adding the non-condensing greenhouse gases to the atmosphere? That is a simple point. But simple does not necessarily mean wrong.
The present article develops the math, which, though not particularly complex, is neither simple nor intuitive. As with previous articles, we shall answer some of the questions raised in comments on the earlier articles. As before, we shall accept ad interim, ad argumentum or ad experimentum all of official climatology except what we can prove to be incorrect.
Let us conduct a simple Gedankenexperiment, running in reverse the model of Lacis et al. (2010), who found that, 50 years after removing all the non-condensing greenhouse gases from the atmosphere, the climate would have settled down to a new equilibrium, giving a slushball or waterbelt Earth with albedo 0.418, implying emission temperature 243.3 K. We shall thus assume ad experimentum that in 1800 there were no greenhouse gases in the atmosphere. For those unfamiliar with the logical modes of argument in scientific discourse, it is not being suggested that there really were no greenhouse gases in 1800.
Lacis found that, only 20 years after removal of the non-condensing greenhouse gases, global mean surface temperature would fall to 253 K. Over the next 30 years it would fall by only 1 K more, to 252 K, or 8.7 K above the emission temperature. Thus, subject to the possibility that the equatorial zone might eventually freeze over, surface temperature in Lacis’ model settled to its new equilibrium after just 50 years.
One question which few opponents in these threads have answered, and none has answered convincingly, is this: What was the source of that additional 8.7 K temperature, given that there were no non-condensing greenhouse gases to drive it? Our answer is that Lacis was implicitly acknowledging the existence of a feedback response to the 243.3 K emission temperature itself – albeit at a value far too small to be realistic. Far too small because, as shown in the previous article, Lacis allocated the 45.1 K difference between the implicit emission temperature of 243.3 K at the specified albedo of 0.418 and today’s global mean surface temperature of 288.4 K (ISCCP, 2018) as follows: Feedback response to emission temperature 252 – 243.3 = 8.7 K; warming directly forced by the naturally-occurring, non-condensing greenhouse gases (288.4 – 252) / 4 = 9.1 K, and, using Lacis’ feedback fraction 0.75, feedback response to warming from the non-condensing greenhouse gases 27.3 K: total 45.1 K. This asymmetric apportionment of the difference between emission temperature and current temperature implies that the 8.7 K feedback response to emission temperature is only 3.6% of 243.3 K, while the 27.3 K feedback response to greenhouse warming is 300% of 9.1 K. Later we shall demonstrate formally that this implausible apportionment is erroneous.
It will be useful to draw a distinction between the pre-industrial position in 1850 (the first year of the HadCRUT series, the earliest global temperature dataset) and the industrial era. We shall assume all global warming before 1850 was natural. That year, surface temperature was about 0.8 K less than today (HadCRUT4) at 287.6 K, or 44.3 K above emission temperature. Lacis’ apportionment of the 44.3 K would thus be 8.7 K, 8.9 K and 26.7 K.
We shall assume that Lacis was right that the directly-forced warming from adding the naturally-occurring, non-condensing greenhouse gases to the air was 8.9 K. Running the experiment in reverse from 1850 allows us to determine the feedback fraction implicit in Lacis’ model after correction to allow for a proper feedback response to emission temperature. Before we do that, let us recall IPCC’s current official list of feedbacks relevant to the derivation of both transient and equilibrium sensitivities:
IPCC’s chosen high-end feedback sum implies Charney sensitivities somewhere between minus infinity and infinity per CO2 doubling. Not a particularly well constrained result after 30 years and hundreds of billions of taxpayers’ dollars. IPCC’s mid-range feedback sum implies a mid-range Charney sensitivity of only 2.2 K, and not the 3.0-3.5 K suggested in previous IPCC reports, nor the 3.3 K in the CMIP3 and CMIP5 ensembles of general-circulation models. No surprise, then, that in 2013, for the first time, IPCC provided no mid-range estimate of Charney sensitivity.
None of the feedbacks listed by IPCC depends for its existence on the presence of any non-condensing greenhouse gas. Therefore, in our world of 1800 without any such gases, all of these feedback processes would be present. To induce a feedback response given the presence of any feedback process, all that is needed is a temperature: i.e., emission temperature. Since feedback processes are present, a feedback response is inevitable.
Emission temperature is dependent on just three quantities: insolation, albedo, and emissivity. Little error arises if emissivity is, as usual, taken as unity. Then, at today’s insolation of 1364.625 Watts per square meter and Lacis’ albedo of 0.418, emission temperature is [1364.625(1 – 0.418) / d / (5.6704 x 10–8)]0.25 = 243.3 K, in accordance with the fundamental equation of radiative transfer, where d, the ratio of the area of the Earth’s spherical surface to that of its great circle, is 4. Likewise, at today’s albedo 0.293, emission temperature would be 255.4 K, the value widely cited in the literature on climate sensitivity.
The reason why official climatology has not hitherto given due weight (or, really, any weight) to the feedback response to emission temperature is that it uses a degenerate form of the zero-dimensional-model equation, ΔTeq = ΔTref / (1 – f ), where equilibrium sensitivity ΔTeq after allowing for feedback is equal to the ratio of reference sensitivity ΔTref to (1 minus the feedback fraction f). The feedback-loop diagram for this equation (below) makes no provision for emission temperature and none, therefore, for any feedback response thereto.
The feedback loop in official climatology’s form of the zero-dimensional-model equation ΔTeq = ΔTref / (1 – f )
Now, this degenerate form of the zero-dimensional-model equation is adequate, if not quite ideal, for deriving equilibrium sensitivities, provided that due allowance has first been made for the feedback response to emission temperature. Yet several commenters find it outrageous that official climatology uses so simple an equation to diagnose the equilibrium sensitivities that the complex general-circulation models might be expected to predict. A few have tried to deny it is used at all. However, Hansen (1984), Schlesinger (1985), IPCC (2007, p. 631 fn.), Roe (2009), Bates (2016) are just a few of the authorities who cite it.
Let us prove by calibration that official climatology’s form of this diagnostic equation, when informed with official inputs, yields the official interval of Charney sensitivities. IPCC (2013, Fig. 9.43) cites Vial et al. (2013) as having diagnosed the CO2 forcing , the Planck parameter and the feedback sum from simulated abrupt 4-fold increases in CO2 concentration in 11 CMIP5 models via the linear-regression method in Gregory (2004). Vial gives the 11 models’ mid-range estimate of the feedback sum as W m–2 K–1, implying , and the bounds of as , i.e. .
The implicit CO2 forcing , in which fast feedbacks were included, was W m–2 compared with the W m–2 in Andrews (2012). Reference sensitivity , taken by Vial as , was above the CMIP5 models’ mid-range estimate . Using these values, official climatology’s version of the zero-dimensional-model equation proves well calibrated, yielding Charney sensitivity on , near-exactly coextensive with several published official intervals from the CMIP3 and CMIP5 climate models (Table 2).
From this successful calibration it follows that, though the equation assumes feedbacks are linear but some feedbacks are nonlinear, it still correctly apportions equilibrium sensitivities between forced warming and feedback response and, in particular, reproduces the interval of Charney sensitivities projected by the CMIP5 models, which do account for nonlinearities. Calibration does not confirm that the models’ value for the feedback fraction or their interval of Charney sensitivities is correct. It does confirm, however, that, at the official values of f, the equation correctly reproduces the official, published Charney-sensitivity predictions from the complex general-circulation models, even though no allowance whatsoever was made for the large feedback response to emission temperature.
Official climatology trains its models by adjusting them until they reproduce past climate. Therefore, the models have been trained to account for the 33 K difference between emission temperature of 255.4 K and today’s surface temperature of 288.4 K. They have assumed that one-quarter to one-third of the 33 K was directly-forced warming from the presence of the naturally-occurring, non-condensing greenhouse gases and the remaining two-thirds to three-quarters was feedback response to that direct warming. Therefore, they have assumed that the feedback fraction was two-thirds to three-quarters of equilibrium sensitivity: i.e., that f was somewhere between 0.67 and 0.75.
As a first step towards making due allowance for the feedback response to emission temperature, official climatology’s version of the zero-dimensional-model equation can be revised to replace the delta input and output signals, indicating mere changes in temperature, with entire or absolute values. Note that the correct form of any equation describing natural occurrences (or any natural law) must be absolute values: the use of deltas is only permissible if the delta-equations are correctly derived from the absolute equation. Accordingly, ΔTeq = ΔTref / (1 – f ) should be Teq = Tref / (1 – f ). The revised feedback loop diagram is below:
After amendment to replace delta inputs and outputs with absolute values, official climatology’s form of the zero-dimensional model equation becomes
Teq = Tref / (1 – f )
To find f where the reference and equilibrium temperatures are known, this revised equation can be rearranged as f = 1 – Tref / Teq. In the reverse Lacis experiment, reference temperature Tref before feedback is the sum of emission temperature TE and the additional temperature ΔTE = 8.9 K that is the direct warming from adding the naturally-occurring, non-condensing greenhouse gases to the air. Thus, Tref = TE + ΔTE = 243.3 + 8.9 = 252.2 K. Equilibrium temperature Teq = 287.6 K is simply the temperature that obtained in 1850, after 50 years of the reverse Lacis experiment. Then f = 1 – Tref / Teq = 1 – 252.2 / 287.6 = 0.123, only a fifth to a sixth of official climatology’s value. The reason for the difference is that, unlike official climatology, we are taking correct account of the feedback response to emission temperature.
Next, how much of the 35.4 K difference between Tref = 252.2 K and Teq = 287.6 K is the feedback response to emission temperature TE = 243.3 K, and how much is the feedback response to the direct greenhouse-gas warming ΔTE = 8.9 K? Simply take the product of each value and f / (1 – f) = 0.14, thus: 243.3 x 0.14 = 34.1 K, and 8.9 x 0.14 = 1.3 K. We prove that this is the correct apportionment by using the standard, mainstream form of the zero-dimensional-model equation that is universal in all dynamical systems except climate. The mainstream equation, unlike the degenerate climate-science form, explicitly separates the input signal (in the climate, the 255.4 K emission temperature) from any amplification (such as the 8.9 K warming from adding the non-condensing greenhouse gases to the atmosphere).
The mainstream zero-dimensional model equation is Teq = Tref μ / (1 – μβ), where Tref is the input signal (here, emission temperature); μ = 1 + ΔTref / Tref is the gain factor representing any amplification of Tref such as that caused by the presence of the naturally-occurring, non-condensing greenhouse gases; β is the feedback fraction; μβ is the feedback factor, equivalent to f in climatology’s current version of the equation; and Teq is equilibrium temperature at re-equilibration of the climate after all feedbacks of sub-decadal duration have acted.
The feedback loop for this corrected form of the zero-dimensional-model equation is below:
The feedback loop diagram for the standard zero-dimensional-model equation
Teq = Tref μ / (1 – μβ)
One advantage of using this mainstream-science form of the zero-dimensional model is that it explicitly and separately accounts for the input signal Tref and for any amplification of it via the gain factor μ in the amplifier, so that it is no longer possible either to ignore or to undervalue either Tref or the feedback response to it that must arise as long as the feedback fraction β is nonzero.
It is proven below that the apportionment of the 35.4 K difference between Tref = 252.2 K and Teq = 287.6 K in 1850 derived earlier in our Gedankenexperiment is in fact the correct apportionment. Starting with the mainstream equation, in due time we introduce the direct or open-loop gain factor μ = 1 + ΔTref / Tref. The feedback factor μβ, the product of the direct or open-loop gain factor μ and the feedback fraction β, has precisely the form that we used in deriving the feedback fraction f as 1 – (243.3 + 8.9) / 287.6 = 0.123, confirming that our apportionment was correct.
Note in passing that in official climatology f is at once the feedback fraction and the feedback factor, since official climatology implicitly (if paradoxically) assumes that the direct or open-loop gain factor μ = 1. In practice, this particular assumption leads official climatology into little error, for the amplification of emission temperature driven by the presence of the non-condensing greenhouse gases is a small fraction of that temperature.
But was it reasonable for us to assume that the 287.6 K temperature in 1850, before Man had exercised any noticeable influence on it, was an equilibrium temperature? Well, yes. We know that in the 168 years since 1850 the world has warmed by only 0.8 K or so, and official climatology attributes all of that warming to Man, not Nature.
Was it reasonable for us to start with Lacis’ implicit emission temperature of 243.3 K, reflecting their specified albedo 0.418 on a waterbelt Earth in the absence of the non-condensing greenhouse gases? Why not start with Pierrehumbert (2011), who said that a snowball Earth would have an albedo 0.6, implying an emission temperature 221.5 K? Let’s do the math. The feedback fraction f = μβ would then be 1 – (221.5 + 8.9) / 287.6 = 0.20.
Thus, from a snowball Earth to 1850, the mean feedback fraction is 0.20; from a waterbelt Earth to 1850, it is 0.12; and at today’s albedo 0.293, implying an emission temperature 255.4 K, it is 1 – (255.4 + 8.9) / 287.6 = 0.08. Which is where we came in at the beginning of this series. For you will notice that, as the great ice sheets melt, the dominance of the surface albedo feedback inexorably diminishes, whereupon the feedback fraction falls over time.
Though the surface albedo feedback may have dominated till now, what about the biggest of all the feedbacks today, the water-vapor feedback? The Clausius-Clapeyron relation implies that the space occupied by the atmosphere may (though not must) carry near-exponentially more water vapor – a greenhouse gas – as it warms. Wentz (2007) found that total column water vapor ought to increase by about 7% per Kelvin of warming. Lacis (2010) allowed for that rate of growth in saying that if one removed the non-condensing greenhouse gases from today’s atmosphere and the temperature fell by 36 K from 288 to 252 K, there would be about 10% of today’s water vapor in the atmosphere: thus, 100% / 1.0736 = 9%.
Specific humidity (g kg–1) at pressure altitudes 300, 6000 and 1000 mb
However, though the increase in column water vapor with warming is thus thought to be exponential, the consequent feedback forcing is approximately logarithmic (just as the direct CO2 forcing is logarithmic). What is more, a substantial fraction of the consequent feedback response is offset by a reduction in the lapse-rate feedback. Accordingly, the water-vapor/lapse-rate feedback response is approximately linear.
Over the period of the NOAA record of specific humidity at three pressure altitudes (above), there was 0.8 K global warming. Therefore, Wentz would have expected an increase of about 5.5% in water vapor. Sure enough, close to the surface, where most of the water vapor is to be found, there was a trend in specific humidity of approximately that value. But the water-vapor feedback response at low altitudes is small because the air is all but saturated already.
However, at altitude, where the air is drier and the only significant warming from additional water vapor might arise, specific humidity actually fell, confirming the non-existence of the predicted tropical mid-troposphere “hot spot” that was supposed to have been driven by increased water vapor. In all, then, there is little evidence to suggest that the temperature response to increased water vapor and correspondingly diminished lapse-rate is non-linear. Other feedbacks are not large enough to make much difference even if they are non-linear.
Our method predicts 0.78 K warming from 1850-2011, and 0.75 K was observed
One commenter here has complained the Planck parameter (the quantity by which a radiative forcing in Watts per square meter is multiplied to convert it to a temperature change) is neither constant nor linear: instead, he says, it is the first derivative of a fourth-power relation, the fundamental equation of radiative transfer. Here, it is necessary to know a little calculus. Adopting the usual harmless simplifying assumption of constant unit emissivity, the first derivative, i.e. the change ΔTref in reference temperature per unit change ΔQ0 in radiative flux density, is simply Tref / (4Q0), which is linear.
A simple approximation to integrate latitudinal variations in the Planck parameter is to take the Schlesinger ratio: i.e., the ratio of surface temperature TS to four times the flux density Q0 = 241.2 Watts per square meter at the emission altitude. At the 255.4 K that would prevail at the surface today without greenhouse gases or feedbacks, the Planck parameter would be 255.4 / (4 x 241.2) = 0.26 Kelvin per Watt per square meter. At today’s 288.4 K surface temperature, the Planck parameter is 288.4 / (4 x 241.2) = 0.30. Not much nonlinearity there.
It is, therefore, reasonable to assume that something like the mean feedback fraction 0.08 derived from the experiment in adding the non-condensing greenhouse gases to the atmosphere will continue to prevail. If so, the equilibrium warming to be expected from the 2.29 Watts per square meter of net industrial-era anthropogenic forcing to 2011 (IPCC, 2013, Fig. SPM.5) will be 2.29 / 3.2 / (1 – 0.08) = 0.78 K. Sure enough, the least-squares linear-regression trend on the HadCRUT4 monthly global mean surface temperature dataset since 1850-2011 (above) shows 0.75 K warming over the period.
But why do the temperature readings from the ARGO bathythermographs indicate a “radiative energy imbalance” suggesting that there is more warming in the pipeline but that the vast heat capacity of the oceans has absorbed it for now?
One possibility is that not all of the global warming since 1850 was anthropogenic. Suppose that the radiative imbalance to 2010 was 0.59 W m–2 (Smith 2015). Warming has thus radiated 2.29 – 0.59 = 1.70 W m–2 (74.2%) to space. Equilibrium warming arising from both anthropogenic and natural forcings to 2011 may thus eventually prove to have been 34.8% greater than the observed 0.75 K industrial-era warming to 2011: i.e., 1.0 K. If 0.78 K of that 1.0 K is anthropogenic, there is nothing to prevent the remaining 0.22 K from having occurred naturally owing to internal variability. This result is actually consistent with the supposed “consensus” proposition that more than half of all recent warming is anthropogenic.
The implication for Charney sensitivity – i.e., equilibrium sensitivity to doubled CO2 concentration – is straightforward. The models find the CO2 forcing to be 3.5 Watts per square meter per doubling. Dividing this by 3.2 to allow for today’s value of the Planck parameter converts that value to a reference sensitivity of 1.1 K. Then Charney sensitivity is 1.1 / (1 – 0.08) = 1.2 K. And that’s the bottom line. Not the 3.3 K mid-range estimate of the CMIP5 models. Not the 11 K imagined by Stern (2006). Just 1.2 K per CO2 doubling. And that is nothing like enough to worry about.
None of the objections raised in response to our result has proven substantial. For instance, Yahoo Answers (even less reliable than Wikipedia) weighed in with the following delightfully fatuous answer to the question “Has Monckton found a fatal error?”
What he does is put forward the following nonsensical argument –
1. If I take the 255.4 K temperature of the earth without greenhouse gases, and I add in the 8K increase with greenhouse gases I get a temperature of 263.4 K.
2. Now, what I’m going to say is say that this total temperature (rather than just the effect of the greenhouse gases) leads to a feedback. And if I use this figure I get a feedback of 1 – (263.4 / 287.6) = 0.08.
And the problem is … how can the temperature of the planet (255.4 K) without greenhouse gases then lead to a feedback? The feedback is due to the gases themselves. You can’t argue that the feedback and hence amplified temperature due to greenhouse gases is actually due to the temperature of the planet without the greenhouse gases! What he’s done is taken the baseline on which the increase and feedback is based, and then circled back to use the baseline as the source of the increase and feedback.
So, I’m afraid it’s total crap …
The error made by Yahoo Answers lies in the false assertion that “the feedback is due to the gases themselves”. No: one must distinguish between the condensing greenhouse gases (a change in the atmospheric burden of water vapor is a feedback process) and the non-condensing greenhouse gases such as CO2 (nearly all changes in the concentration of the non-condensing gases are forcings). All of the feedback processes listed in Table 1 would be present even in the absence of any of the non-condensing greenhouse gases.
Another objection is that perhaps official climatology makes full allowance for the feedback response to emission temperature after all. That objection may be swiftly dealt with. Here is the typically inspissate and obscurantist definition of a “climate feedback” in IPCC (2013):
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.
IPCC’s definition thus explicitly excludes any possibility of a feedback response to a pre-existing temperature, such as the 255.4 K emission temperature that would prevail at the surface in the absence of any greenhouse gases or feedbacks. It was for this reason that Roy Spencer thought we must be wrong.
Our simple point remains: how can an inanimate feedback process know how to distinguish between the input emission of temperature of 255 K and a further 9 K of temperature arising from the addition of the non-condensing greenhouse gases to the atmospheric mix? How can it know it should react less to the former than to the latter, or (if IPCC’s definition is followed) not at all to the former and extravagantly to the latter? In the end, despite some valiant attempts by true-believers to complicate matters, our point is as simple – and in our submission as unanswerable – as that.