Willis and I walk the Planck

By Christopher Monckton of Brenchley

I do apologize for not having replied sooner to my friend the irrepressible, irascible, highly improbable but always fascinating Willis Eschenbach, who on August 15 had commented on a brace of earlier postings by me on the vexed question of climate sensitivity.

The delay is because my lovely wife and I are on a two-week trip to the thrusting new Scotland of the ghastly totalitarians who call themselves the Scottish National Party. We had left our beloved Scotland five years ago when we had sensed the advance of the legalists – as the early Chinese philosophers would have called today’s totalitarians. We are what the Chinese would have called Confucians – in today’s money, libertarians.

How said it is to see the Scotland we left just five years ago in such rampant and almost entirely unreported decline. Even in Perth, our old and once prosperous county town, the thriving shops have largely gone, to be replaced by dismal bingo-halls, desperate charity shops and boarded-up windows.

The cottage where we were unwise enough to lodge during our first week’s visit cost us well north of $1000 for the week, and the wretches who let it to us did not provide free electricity or even logs for the fire. The place was filthy; the oven unusable; the wood-burning stove so clogged with clag that one could not see through the glass to where the fire within would have been if there had been any logs; the gutters not maintained; the water not even basically filtered to remove the lumps of peat that turned my white shirts brown. And the internet? They said that if we stood in the kitchen corner we might occasionally get a flicker of a signal. Well, we didn’t get one.

In Paris last December, while playing the piano for the late and much missed Bob Carter in the swank foyer of the grandest of grand hotels on the Champs-Elysees, I mentioned to him that we were by no means the only ones who had left Scotland. Had I been younger, I said, I’d have stood and fought. As it was, a tide of talent and brains and wealth was pouring southward; businesses were closing down all over the boarded-up shop; the oil price had collapsed; and everyone who was anyone was getting out.

Bob said, “Monckton, you’re exaggerating. And I’m going to prove it.” He got out his cellphone and telephoned a friend in Aberdeen who employed 400 people there. “Is it true,” he asked, “that there is an unreported exodus from Scotland?”

“I can’t speak for the whole of Scotland,” said Bob’s friend, “But I will say this. I and all 400 of my employees are leaving just as soon as we can get out.”

Which explains why there was no internet. The notion of providing a service has now largely vanished from Scotland. Inferentially, the signal from furth of Shangri-La did not reach our corner of the damp cottage kitchen because the Amalgamated Union of Semaphore Flag-Wavers and Mountain-Top Beacon Fettlers was on strike. Again.

So to Willis’ posting.

Willis thought I was wrong (Wrong? Moi?) about the value of the pre-feedback climate-sensitivity parameter, widely known in the climate literature (see e.g. Roe, 2009) as λ₀.

Misleadingly, Willis refers to this “Planck parameter” as a “feedback”. Properly understood, it is nothing of the kind: for, as the equation that I had illustrated in my previous postings demonstrated, its role in determining climate sensitivity – and that was the role in which I had cast it – is manifestly distinct from that of any true feedback.

Willis says: “The Planck feedback is how much the outgoing long-wave radiation of the globe increases per degree of increased temperature.” It is much better understood the other way about, for the models use its reciprocal, the Planck parameter, to convert Watts per square meter of long-wave radiation change (i.e., of forcing) to Kelvin of temperature change (i.e., climate sensitivity). See the interesting discussion in Roe (2009) on this point.

The Planck parameter, which I shall accordingly denominate hereafter in Kelvin of temperature change per Watt per square meter of radiative flux-density change, occurs twice in the official climate-sensitivity equation.

First, at the pre-feedback stage, the Planck parameter is the constant of proportionality that converts any change in long-wave radiation as a result of a radiative forcing such as atmospheric CO₂ enrichment into a corresponding change in temperature.

Secondly, the Planck parameter acts in exactly the same way on temperature feedbacks. Feedbacks are denominated in Watts per square meter per Kelvin of temperature change arising from the original, direct forcing. The product of the Planck parameter (in purple) and the sum of these feedbacks (in bright blue) is the unitless temperature-feedback factor f (in pink) in my illuminated presentation of the official climate-sensitivity equation.

The value of the Planck parameter is, therefore, of paramount importance. And Willis, who is prone to rush to the data (which, to be fair, are usually not a bad place to start), rushed to the data and determined the value of the Planck parameter not as the 0.313 Kelvin per Watt per square meter that I (supported by IPCC and dozens of scientific papers and models I could name) had asserted, but a mere 0.2 Kelvin per Watt per square meter.

How come this discrepancy?

Simple. Willis, in his posting, made the same mistake that I had myself made in the very first article I had written on climate sensitivity, which had appeared ten years ago all over the front page of the Weekend section of the London Sunday Telegraph and had been so popular with readers that it crashed the Telegraph website for the first and only time in its history, attracting the then-unheard-of hit-rate of 127,000 hits in two hours at midnight on a Sunday morning. By that metric, it was the most popular article the Telegraph group had ever published.

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 mistake that Willis (and, at that time, I) had made was to perform the calculation to determine the Planck sensitivity at the Earth’s surface and not, as it should be performed for climate-sensitivity studies, at the Planck emission surface, whose mean pressure altitude is about 300 hPa up in the mid-troposphere.

The Planck emission surface is, by definition, the locus of all points of least altitude at which incoming and outgoing radiation are equal in the atmospheric columns that may be thought of as subsisting above all points on the Earth’s surface.

This strange surface is the surface from which satellites perceive outgoing radiation from the Earth to emanate. It is – again by definition – one optical depth down into the atmosphere as seen from above.

And it is at this emission surface, and not at the Earth’s hard-deck surface, that the Planck parameter falls to be determined.

Here is how it is done. There is really very little argument about the value of the Planck parameter, for its derivation is so very straightforward.

Begin with the data (Willis will like that bit). The SORCE/TIM data show that the mean total solar irradiance is about 1361 Watts per square meter, and all datasets are within a few Watts per square meter of this value, so I shall use the SORCE/TIM value.

The Earth presents a disk-shaped cross-section to the incoming radiation, but its surface is a rotating sphere. So it is necessary to divide the total solar irradiance by 4, which is the ratio of the surface area of a sphere to that of a disk of equal radius.

Next, one must allow for albedo. The Earth (or, in particular, the clouds, which account for some 97% of its albedo) reflect about 30% of all incoming solar radiation harmlessly straight back into space. So the mean flux density at the Earth’s emission altitude is 1361 (1 – 0.3) / 4 = 238.2 Watts per square meter.

Now it is time to determine the mean emission temperature represented by that radiation of 238.2 Watts per square meter. This is done by using one of the very few proven results in the generally slippery subject of climatology – the fundamental equation of radiative transfer.

The equation states that the radiative flux at the emission surface of a celestial body is equal to the product of just three values: the emissivity of that surface, the Stefan-Boltzmann constant and the fourth power of temperature.

Since we know the radiation at the Earth’s emission surface, and we know that after allowance for albedo the emissivity of that surface is unity, and we know the Stefan-Boltzmann constant is reassuringly constant at 0.000000056704 Watts per square meter per Kelvin to the fourth power, it is a simple matter to deduce the one unknown quantity in the equation: the Earth’s emission temperature, which turns out to be 254.6 Kelvin, or around 34 Kelvin cooler than the hard-deck surface where we live and move and have our being.

To find out the relationship between any change in radiative flux density at the emission surface and any consequent change in the temperature at that surface, it is necessary only to take the first derivative of the fundamental equation of radiative transfer.

It is not always appreciated that, provided that one expresses the derivative in terms of both temperature and flux density, the relation between radiation change and temperature change is linear, even though the derivative comes from a fourth-power relation.

Here is the math:

K W^–1 m²_.

One final adjustment is needed, and, to verify IPCC’s value, some years ago I obtained from John Christy a datafile containing 30 years’ temperature-anomaly data for the mid-troposphere. Using these data (Willis would be pleased again), I was able to determine the Hölder coefficient from the integration of latitudinal values for λ₀ using equialtitudinal latitudinal frusta, for are not frusta that are equaialtitudinal also conveniently equiareal? [Hint: yes, they are].

The bottom line: the product of the Hölder coefficient 7/6 (which allows for the fact that a sum of latitudinally-derived fourth powers, for instance, is not the same as the fourth power of a sum) and the first differential obtained by taking the derivative above gives a very good approximation to the current value of the Planck parameter λ₀, namely 0.313 K W^–1 m².

Phew!

Can the value of the Planck parameter vary? Yes, if insolation varies, and yes, if albedo varies. But, since the solar “constant” is near-invariant, and since the albedo is unlikely to change much even if major ice losses eventually occur, lambda-zero will continue to be at or close to 0.313 K W^–1 m² for the foreseeable future.

With respect, therefore, Willis was infelicitous in referring to the Planck parameter as a “feedback”, for it is unlike any true feedback; he was incorrect (as I had once been) in attempting to determine it at the hard-deck surface rather than the emission surface of the Earth; he was accordingly incorrect (as I had once been) in determining its value to be of order 0.2 Kelvin per Watt per square meter; he was incorrect in imagining the Planck parameter to be non-linear (I knew enough calculus not to fall for that one); and he was incorrect in imagining that its value had been determined without regard to latitudinal non-linearities (I do more homework than I usually show in these columns for general family entertainment).

Apart from that, Mrs Lincoln, how did you enjoy the play?

But let us end with a richly-deserved compliment to Willis. Like me, he is largely an autodidact. Like me, he makes mistakes. And this time I am in no position to crow: for the mistakes he has made are the mistakes I had once made myself.

Above all, like me he is interested enough to ask questions – usually very good questions – and to do his very best to find the answers. To him, as to me, science is a matter not of belief but of diligent, disciplined inquiry. It is this passionate curiosity that unites us, and marks us out from the totalitarian true-believers who are wrecking Scotland and have done their best to wreck science too. To them, and not to him, I award the accolade “Thick as two short Plancks”.