1 K or not 1 K? That is the question

By Christopher Monckton of Brenchley

I am very grateful for the many thoughtful postings in response to my outline of the fundamental theoretical upper bound of little more than 1.2 K on climate sensitivity imposed by the process-engineering theory of maintaining the stability of an object on which feedbacks operate. Here are some answers to points raised by correspondents.

Iskandar says, “None of these feedbacks or forcings are ever given in the form of a formula.” In fact, there are functions for the forcings arising from each of the principal species of greenhouse gas: they are tabulated in Myhre et al., 1998, and cited with approval in IPCC (2001. 2007). However, Iskandar is right about temperature feedbacks. Here, the nearest thing to a formula for a feedback is the Clausius-Clapeyron relation, which states that the space occupied by the atmosphere is capable of carrying near-exponentially more water vapor as it warms. However, as Paltridge et al. (2009) have indicated, merely because the atmosphere can carry more water vapor there is no certainty that it does. The IPCC’s values for this and other feedbacks are questionable. For instance, Spencer and Braswell (2010, 2011, pace Dessler, 2010, 2011) have challenged the IPCC’s estimate of the cloud feedback. They find it as strongly negative (attenuating the warming that triggers it) as the IPCC finds it strongly positive (amplifying the original warming), implying a climate sensitivity of less than 1 K. Since feedbacks account for almost two-thirds of all warming in the IPCC’s method, and since it is extremely difficult to measure – still less to provide a formula for – the values of individual temperature feedbacks, an effort such as mine to identify a constraint on the magnitude of all feedbacks taken together is at least worth trying.

Doug says we cannot be sure when the dolomitic rocks were formed. What is certain, however, according to Professor Ian Plimer, who gave me the information, is that they cannot form unless the partial pressure of CO2 above the ocean in which they form is 30%, compared with today’s 0.04%. Yet, during the long era when CO2 concentrations were that high, glaciers came and went, twice, at sea level, and at the equator. Even allowing for the fact that the Sun was a little fainter then, and that the Earth’s albedo was higher, the presence of those glaciers where there are none today does raise some questions about the forcing effect of very high CO2 concentrations, and, a fortiori, about the forcing effect of today’s mere trace concentration. However, in general Doug’s point is right: it is unwise to put too much weight on results from the paleoclimate, particularly when there is so much scientific dispute about the results from today’s climate that we can measure directly.

Dirk H and the inimitable Willis Eschenbach, whose fascinating contributions to this column should surely be collected and published as a best-seller, point out that I am treating feedbacks as linear when some of them are non-linear. For the math underlying non-linear feedbacks, which would have been too lengthy to include in my posting, see e.g. Roe (2009). Roe’s teacher was Dick Lindzen, who is justifiably proud of him. However, for the purpose of the present argument, it matters not whether feedbacks are linear or non-linear: what matters is the sum total of feedbacks as they are in our own time, which is multiplied by the Planck parameter (of which more later) to yield the closed-loop gain whose upper bound was the focus of my posting. Of course I agree with Willis that the non-linearity of many feedbacks, not to mention that all or nearly all of them cannot be measured directly, makes solving the climate-sensitivity equation difficult. But, again, that is why I have tried the approach of examining a powerful theoretical constraint on the absolute magnitude of the feedback-sum. Since the loop gain in the climate object cannot exceed 0.1 (at maximum) without rendering the climate so prone to instability that runaway feedbacks that have not occurred in the past would be very likely to have occurred, the maximum feedback sum before mutual amplification cannot exceed 0.32: yet the IPCC’s implicit central estimate of the feedback sum is 2.81.

Roger Knights rightly takes me to task for a yob’s comma that should not have been present in my posting. I apologize. He also challenges my use of the word “species” for the various types of greenhouse gas: but the word “species” is regularly used by the eminent professors of climatology at whose feet I have sat.

R. de Haan cites an author whose opinion is that warming back-radiation returned from the atmosphere back to the surface and the idea that a cooler system can warm a warmer system are “unphysical concepts”. I know that the manufacturers of some infra-red detectors say the detectors do not measure back-radiation but something else: however, both Mr. de Haan’s points are based on a common misconception about what the admittedly badly-named “greenhouse effect” is. The brilliant Chris Essex explains it thus: when outgoing radiation in the right wavelengths of the near-infrared meets a molecule of a greenhouse gas such as CO2, it sets up a quantum resonance in the gas molecule, turning it into a miniature radiator. This beautifully clear analogy, when I recently used it in a presentation in New Zealand, won the support of two professors of climatology in the audience. The little radiators that the outgoing radiation turns on are not, of course, restricted only to radiating outwards to space. They radiate in all directions, including downwards – and that is before we take into account non-radiative transports such as subsidence and precipitation that bring some of that radiation down to Earth. So even the IPCC, for all its faults, is not (in this respect, at any rate) repealing the laws of thermodynamics by allowing a cooler system to warm a warmer system, which indeed would be an unphysical concept.

Gary Smith politely raised the question whether the apparently sharp ups and downs in the paleoclimate temperature indicated strongly-positive feedbacks. With respect, the answer is No, for two reasons. First, the graph I used was inevitably compressed: in fact, most of the temperature changes in that graph took place over hundreds of thousands or even millions of years. Secondly, it is the maximum variance either side of the long-run mean, not the superficially-apparent wildness of the variances within the mean, that establishes whether or not there is a constraint on the maximum net-positivity of temperature feedbacks.

Nick Stokes asked where the limiting value 0.1 for the closed-loop gain in the climate object came from. It is about an order of magnitude above the usual design limit for net-positive feedbacks in electronic circuits that are not intended to experience runaway feedbacks or to oscillate either side of the singularity in the feedback-amplification equation, which occurs where the loop gain is unity.

David Hoffer wondered what evidence the IPCC had for assuming a linear rise in global temperature over the 21^st century given that the radiative forcing from CO2 increases only at a logarithmic (i.e. sub-linear) rate. The IPCC pretends that all six of its “emissions scenarios” are to be given equal weight, but its own preference for the A2 scenario is clear, particularly in the relevant chapter of its 2007 report (ch. 10). See, in particular, fig. 10.26, which shows an exponential rise in both CO2 and temperature, when one might have expected the logarithmicity of the CO2 increase to cancel the exponentiality of the temperature increase. However, on the A2 scenario it is only the anthropogenic fraction of the CO2 concentration that is increased exponentially, and this has the paradoxical effect of making temperature rise near-exponentially too – but only if one assumes the very high climate sensitivity that is impossible given the fundamental constraint on the net-positivity of temperature feedbacks.

DR asks whether anyone has ever actually replicated experimentally the greenhouse effect mentioned by Arrhenius, who in 1895/6 first calculated how much warming a doubling of CO2 concentration would cause. Yes, the greenhouse effect was first demonstrated empirically by John Tyndale at the Royal Institution, London (just round the corner from my club) as far back as 1859. His apparatus can still be seen there. The experiment is quite easily replicated, so we know (even if the SB equation and the existence of a readily-measurable temperature lapse-rate with altitude did not tell us) that the greenhouse effect is real. The real debate is not on whether there is a greenhouse effect (there is), but on how much warming our rather small perturbation of the atmosphere with additional concentrations of greenhouse gases will cause (not a lot).

Werner Brozek asks whether the quite small variations in global surface temperature either side of the billion-year mean indicate that “tipping-points” do not exist. In mathematics and physics the term “tipping-point” is really only used by those wanting to make a political point, usually from a climate-extremist position. The old mathematical term of art, still used by many, was “phase-transition”: now we should usually talk of a “bifurcation” in the evolution of the object under consideration. Since the climate object is mathematically-chaotic (IPCC, 2001, para. 14.2.2.2; Giorgi, 2005; Lorenz, 1963), bifurcations will of course occur: indeed any sufficiently rare extreme-weather event may be a bifurcation. We know that very extreme things can suddenly happen in the climate. For instance, at the end of the Younger Dryas cooling period that brought the last Ice Age to an end, temperatures in Antarctica as inferred from variations in the ratios of different isotopes of oxygen in air trapped in layers under the ice, rose by 5 K (9 F) in just three years. “Now, that,” as Ian Plimer likes to say in his lectures, “is climate change!”

But the idea that our very small perturbation in temperature will somehow cause more bifurcations is not warranted by the underlying mathematics of chaos theory. In my own lectures I often illustrate this with a spectacular picture drawn on the Argand plane by a very simple chaotic function, the Mandelbrot fractal function. The starting and ending values for the pixels at top right and bottom left respectively are identical to 12 digits of precision; yet the digits beyond 12 are enough to produce multiple highly-visible bifurcations.

And we know that some forms of extreme weather are likely to become rarer if the world warms. Much – though not all – extreme weather depends not upon absolute temperature but upon differentials in temperature between one altitude or latitude and another. These differentials tend to get smaller as the world warms, so that outside the tropics (and arguably in the tropics too) there will probably be fewer storms.

Roy Clark says there is no such thing as equilibrium in the climate. No, but that does not stop us from trying to do the sums on the assumption of the absence of any perturbation (the equilibrium assumption). Like the square root of -1, it doesn’t really exist, but it is useful to pretend ad argumentum that it might.

Legatus raised a fascinating point about the measurements of ambient radiation that observatories around the world make so that they can calibrate their delicate, heat-sensitive telescopes. He says those measurements show no increase in radiation at the surface (or, rather, on the mountain-tops where most of the telescopes are). However, it is not the surface radiation but the radiation at the top of the atmosphere (or, rather, at the characteristic-emission altitude about 5 km above sea level) that is relevant: and that is 239.4 Watts (no relation) per square meter, by definition, because the characteristic-emission altitude (the outstanding Dick Lindzen’s name for it) is that altitude at which outgoing and incoming fluxes of radiation balance. It is also at that altitude, one optical depth down into the atmosphere, that satellites “see” the radiation coming up into space from the Earth/atmosphere system. Now, as we add greenhouse gases to the atmosphere and cause warming, that altitude will rise a little; and, because the atmosphere contains greenhouse gases and, therefore, its temperature is not uniform, consequent maintenance of the temperature lapse-rate of about 6.5 K/km of altitude will ensure that the surface warms as a result. Since the altitude of the characteristic-emission level varies by day and by night, by latitude, etc., it is impossible to measure directly how it has changed or even where it is.

Of course, it is at the characteristic-emission altitude, and not – repeat not – at the Earth’s surface that the Planck parameter should be derived. So let me do just that. Incoming radiation is, say, 1368 Watts per square meter. However, the Earth presents itself to that radiation as a disk but is actually a sphere, so we divide the radiation by 4 to allow for the ratio of the surface areas of disk and sphere. That gives 342 Watts per square meter. However, 30% of the Sun’s radiation is reflected harmlessly back to space by clouds, snow, sparkling sea surfaces, my lovely wife’s smile, etc., so the flux of relevant radiation at the characteristic-emission altitude is 342(1 – 0.3) = 239.4 Watts per square meter.

From this value, we can calculate the Earth’s characteristic-emission temperature directly without even having to measure it (which is just as well, because measuring even surface temperature is problematic). We use the fundamental equation of radiative transfer, the only equation to be named after a Slovene. Stefan found the equation by empirical methods and, a decade or so later, his Austrian pupil Ludwig Boltzmann proved it theoretically by reference to Planck’s blackbody law (hence the name “Planck parameter”, engagingly mis-spelled “plank” by one blogger.

The equation says that radiative flux is equal to the emissivity of the characteristic-emission surface (which we can take as unity without much error when thinking about long-wave radiation), times the Stefan-Boltzmann constant 5.67 x 10^–8 Watts per square meter per Kelvin to the fourth power, times temperature in Kelvin to the fourth power. So characteristic-emission temperature is equal to the flux divided by the emissivity and by the Stefan-Boltzmann constant, all to the power 1/4.: thus, [239.4 / (1 x 5.67 x 10^–8)]^¼ = 254.9 K or thereby.

Any mathematician taking a glance at this equation will at once notice that one needs quite a large change in radiative flux to achieve a very small change in temperature. To find out how small, one takes the first differential of the equation, which (assuming emissivity to be constant) is simply the temperature divided by four times the flux: so, 254.9 / (4 x 239.4) = 0.2662 Kelvin per Watt per square meter. However, the IPCC (2007, p. 631, footnote) takes 0.3125 and, in its usual exasperating way, without explaining why. So a couple of weeks ago I asked Roy Spencer and John Christy for 30 years of latitudinally-distributed surface temperature data and spent a weekend calculating the Planck parameter at the characteristic-emission altitude for each of 67 zones of latitude, allowing for latitudinal variations in insolation and adjusting for variations in the surface areas of the zones. My answer, based on the equinoxes and admittedly ignoring seasonal variations in the zenith angles of the Sun at each latitude, was 0.316. So I’ve checked, and the IPCC has the Planck parameter right. Therefore, it is of course the IPCC’s value that I used in my calculations in my commentary for Remote Sensing, except in one place.

Kiehl & Trenberth (1997) publish a celebrated Earth/atmosphere energy-budget diagram in which they show 390 Watts per square meter of outgoing radiative flux from the surface, and state that this is the “blackbody” value. From this, we know that – contrary to the intriguing suggestion made by Legatus that one should simply measure it – they did not attempt to find this value by measurement. Instead, they were taking surface emissivity as unity (for that is what defines a blackbody), and calculating the outgoing flux using the Stefan-Boltzmann equation. The surface temperature, which we can measure (albeit with some uncertainty) is 288 K. So, in effect, Kiehl and Trenberth are saying that they used the SB equation at the Earth’s surface to determine the outgoing surface flux, thus: 1 x 5.67 x 10^–8 x 288^4 = 390.1 Watts per square meter.

Two problems with this. First, the equation holds good only at the characteristic-emission altitude, and not at the surface. That is why, once I had satisfied myself that the IPCC’s value at that altitude was correct, I said in my commentary for Remote Sensing that the IPCC’s value was correct, and I am surprised to find that a blogger had tried to leave her readers with a quite different impression even after I had clarified this specific point to her.

Secondly, since Kiehl and Trenberth are using the Stefan-Boltzmann equation at the surface in order to obtain their imagined (and perhaps imaginary) outgoing flux of 390 Watts per square meter, it is of course legitimate to take the surface differential of the equation that they themselves imply that they had used, for in that we we can determine the implicit Planck parameter in their diagram. This is simply done: 288 / (4 x 390) = 0.1846 Kelvin per Watt per square meter. Strictly speaking, one should also add the non-radiative transports of 78 Watts per square meter for evapo-transpiration and 24 for thermal convection (see Kimoto, 2009, for a discussion) to the 390 Watts per square meter of radiative flux, reducing Kiehl and Trenberth’s implicit Planck parameter from 0.18 to 0.15. Either 0.15 or 0.18 gives a climate sensitivity ~1 K. So the Planck parameter I derived at this point in my commentary, of course, not the correct one: nor is it “Monckton’s” Planck parameter, and the blogger who said it was had been plainly told all that I have told you, though in a rather more compressed form because she had indicated she was familiar with differential calculus. It is not Monckton’s Planck parameter, nor even Planck’s Planck parameter, and it is certainly not a plank parameter – but it is Kiehl & Trenberth’s Planck parameter. If they were right (and, of course, I was explicit in using the conditional in my commentary to indicate, in the politest possible way, that they were not), then, like it or not, they were implying a climate sensitivity a great deal lower than they had perhaps realized – in fact a sensitivity of around 1 K. I do regret that a quite unnecessary mountain has been made out of this surely simple little molehill – just one of more than a dozen points in a wide-ranging commentary.

And just to confirm that it should really have been obvious to everyone that the IPCC’s value of the Planck parameter is my value, I gave that value as the correct one both in my commentary and in my recent blog posting on the fundamental constraint on feedback loop gain. You will find it, with its derivation, right at the beginning of that posting, and encapsulated in Eq. (3).

Thank you all again for your interest. This discussion has generally been on a far higher plane than is usual with climate discussions. I hope that these further points in answer to commentators will be helpful.