My commentary written for Remote Sensing on the empirical determination of climate sensitivity, published by the splendid Anthony Watts some days ago, has aroused a great deal of interest among his multitudes of readers. It is circulating among climate scientists on both sides of the debate. Several of Anthony’s readers have taken the trouble to make some helpful comments. Since some of these are buried among the usual debates between trolls on how awful I am, and others were kindly communicated privately, I have asked Anthony to allow me, first and foremost, to thank those readers who have been constructive with their comments, and to allow his readers the chance to share the comments I have received.
Joel Shore pointed out that Schwartz, whose paper of 2007 I had cited as finding climate sensitivity to be ~1 K, wrote a second paper in 2008 finding it close to 2 K. Shore assumed I had seen but suppressed the second paper. By now, most of Anthony’s readers will perhaps think less ungenerously of me than that. The new .pdf version of the commentary, available from Anthony’s website (here), omits both Schwartz papers: but they will be included in a fuller version of the argument in due course, along with other papers which use observation and measurement, rather than mere modeling, to determine climate sensitivity.
Professor Michael Asten of Monash University helpfully provided a proper reference in the reviewed literature for Christopher Scotese’s 1999 paper reconstructing mean global surface temperatures from the Cambrian Era to the present. This, too, has been incorporated into the new .pdf.
Professor Asten also supplied a copy of a paper by David Douglass and John Christy, published in that vital outlet for truth Energy & Environment in 2009, and concluding on the basis of recent temperature trends that feedbacks were not likely to be net-positive, implying climate sensitivity ~1 K. I shall certainly be including that paper and several others in the final version of the full-length paper that underlies the commentary published by Anthony. This paper is now in draft and I should be happy to send it to any interested reader who emails firstname.lastname@example.org.
A regular critic, Lucia Liljegren was, as all too often before, eager to attack my calculations – she erred in publishing a denial that I sent her a reference that I can prove she received; and not factually accurate in blogging that “Monckton’s” Planck parameter was “pulled out of a hat” when I had shown her that in my commentary I had accepted the IPCC’s value as correct. She was misleading her readers in not telling them that the “out-of-a-hat” relationship she complains of is one which Kiehl and Trenberth (1997) had assumed, with a small variation (their implicit λ0 is 0.18 rather than the 0.15 I derived from their paper via Kimoto, 2009); and selective in not passing on that I had told her they were wrong to assume that a blackbody relationship between flux and temperature holds at the surface (if it did, as my commentary said, it would imply a climate sensitivity ~1 K).
A troll (commenter on WUWT) said I had “fabricated” the forcing function for CO2. When I pointed out that I had obtained it from Myhre et al. (1998), cited with approval in IPCC (2001, 2007), he whined at being called a troll (so don’t accuse me of “fabricating” stuff, then, particularly when I have taken care to cite multiple sources, none of which you were able to challenge) and dug himself further in by alleging that the IPCC had also “fabricated” the CO2 forcing function. No: the IPCC got it from Myhre et al., who in turn derived it by inter-comparison between three models. I didn’t and don’t warrant that the CO2 forcing function is right: that is above my pay-grade. However, Chris Essex, the lively mathematician who did some of the earliest spectral-line modeling of the CO2 forcing effect, confirms that Myhre and the IPCC are right to state that the function is a logarithmic one. Therefore, until I have evidence that it is wrong, I shall continue to use it in my calculations.
Another troll said – as usual, without providing any evidence – that I had mis-stated the result from process engineering that provides a decisive (and low) upper bound to climate sensitivity. In fact, the result came from a process engineer, Dr. David Evans, who is one of the finest intuitive mathematicians I have met. He spent much of his early career designing and building electrical circuitry and cannot, therefore, fairly be accused of not knowing what he is talking about. Since the resulting fundamental upper limit to climate sensitivity is as low as 1.2 K, I thought readers might be interested to have a fuller account of it, which is very substantially the work of David Evans. It is posted below this note.
Hereward Corley pointed out that the reference to Shaviv (2008) should have been Shaviv (2005). Nir Shaviv – another genius of a mathematician – had originally sent me the paper saying it was from 2008, but the version he sent was an undated pre-publication copy. Mr. Corley also kindly supplied half a dozen further papers that determine climate sensitivity empirically. Most of the papers find it low, and all find it below the IPCC’s estimates. The papers are Chylek & Lohman (2008); Douglass & Knox (2005); Gregory et al. (2002); Hoffert & Covey (1992); Idso (1998); and Loehle & Scafetta (2011).
I should be most grateful if readers would be kind enough to draw my attention to any further papers that determine climate sensitivity by empirical methods rather than by the use of general-circulation models. I don’t mind what answers the papers come to, but I only want those that attempted to reach the answer by measurement, observation, and the application of established theory to the results.
Many thanks again to all of you for your interest and assistance. Too many of the peer-reviewed journals are no longer professional enough or unprejudiced enough to publish anything that questions the new State religion of supposedly catastrophic manmade global warming. Remote Sensing, for instance has still not had the courtesy to acknowledge the commentary I sent. Since the editors of the learned journals seem to have abdicated their role as impartial philosopher-kings, WattsUpWithThat is now the place where (in between the whining and whiffling and waffling of the trolls) true science is done.
The fundamental constraint on climate sensitivity
A fundamental constraint rules out strongly net-positive temperature feedbacks acting to amplify warming triggered by emissions of greenhouse gases, with the startling result that climate sensitivity cannot much exceed 1.2 K.
Sensitivity to doubled CO2 concentration is the product of three parameters (Eq. 1):
- the radiative forcing ΔF2x = 5.35 ln 2 = 3.708 W m–2 at CO2 doubling (Eq. 2), from the function in Myhre et al. (1998) and IPCC (2001, 2007);
- the Planck zero-feedback climate sensitivity parameter λ0 = 0.3125 K W–1 m2 (Eq. 3), equivalent to the first differential of the fundamental equation of radiative transfer in terms of mean emission temperature TE and the corresponding flux FE at the characteristic-emission altitude (CEA, one optical depth down into the atmosphere, where incoming and outgoing fluxes are identical), augmented by approximately one-sixth to allow for latitudinal variation (IPCC, 2007, p. 631 fn.);
- the overall feedback gain factor G (Eq. 4), equivalent, where feedbacks are assumed linear as here, to (1 – g)–1, where the feedback loop gain g is the product of λ0 and the sum f of all unamplified temperature feedbacks f1, f2, … fn, such that the final or post-feedback climate sensitivity parameter λ is the product of λ0 and G.
The values of the first two of the three parameters whose product is climate sensitivity are known (Eqs. 2-3). The general-circulation models, following pioneering authors such as Hansen (1984), assume that the feedbacks acting upon the climate object are strongly net-positive (G 1: the IPCC’s implicit central estimate is G = 2.81). In practice, however, neither individual temperature feedbacks nor their sum can be directly measured; nor can feedbacks be readily distinguished from forcings (Spencer & Braswell, 2010, 2011; but see Dessler, 2010, 2011).
Temperature feedbacks – in effect, forcings that occur because a temperature change has triggered them – are the greatest of the many uncertainties that complicate the determination of climate sensitivity. The methodology that the models adopt was first considered in detail by Bode (1945) and is encapsulated at its simplest, assuming all feedbacks are linear, in Eq. (4). Models attempt to determine the value of each distinct positive (temperature-amplifying) and negative (temperature-attenuating) feedback in Watts per square meter per Kelvin of original warming. The feedbacks f1, f2, … fn are then summed and mutually amplified (Eq. 4).
Fig. 1 schematizes the feedback loop:
Figure 1. A forcing ΔF is input (top left) by multiplication to the final sensitivity parameter λ = λ0G, where g = λ0f = 0.645 is the IPCC’s implicit central estimate of the loop gain and G = (1 – g)–1 = 2.813 [not shown] is the overall gain factor: i.e., the factor by which the temperature change T0 = ΔF λ0 triggered by the original forcing is multiplied to yield the output final climate sensitivity ΔT = ΔF λ = ΔF λ0 G (top right). To generate λ = λ0 G, the feedbacks f1, f2, … fn, summing to f, are mutually amplified via Eq. (4). Stated values of λ0, f, g, G, and λare those implicit in the IPCC’s central estimate ΔT2x = 3.26 K (2007, p. 798, Box 10.2) in response to ΔF2x = 5.35 ln 2 = 3.708 W m–2. Values for individual feedbacks f1-f4 are taken from Soden & Held (2006). (Author’s diagram from a drawing by Dr. David Evans).
The modelers’ attempts to identify and aggregate individual temperature feedbacks, while understandable, do not overcome the difficulties in distinguishing feedbacks from forcings or even from each other, or in determining the effect of overlaps between them. The methodology’s chief drawback, however, is that in concentrating on individual rather than aggregate feedbacks it overlooks a fundamental physical constraint on the magnitude of the feedback loop gain g in Eq. (4).
Paleoclimate studies indicate that in the past billion years the Earth’s absolute global mean surface temperature has not varied by more than 3% (~8 K) either side of the 750-million-year mean (Fig. 2):
Figure 2. Global mean surface temperature over the past 750 million years, reconstructed by Scotese (1999), showing variations not exceeding 8 K (<3%) either side of the 291 K (18 °C) mean.
Consistent with Scotese’s result, Zachos et al. (2001), reviewing detailed evidence from deep-sea sediment cores, concluded that in the past 65 Ma the greatest departure from the long-run mean was an increase of 8 K at the Poles, and less elsewhere, during the late Paleocene thermal maximum 55 Ma BP.
While even a 3% variation either side of the long-run mean causes ice ages at one era and hothouse conditions at another, in absolute terms the temperature homeostasis of the climate object is formidable. At no point in the geologically recent history of the planet has a runaway warming occurred. The Earth’s temperature stability raises the question what is the maximum feedback loop gain consistent with the long-term maintenance of stability in an object upon which feedbacks operate.
The IPCC’s method of determining temperature feedbacks is explicitly founded on the feedback-amplification equation (Eq. 4, and see Hansen, 1984) discussed by Bode (1945) in connection with the prevention of feedback-induced failure in electronic circuits. A discussion of the methods adopted by process engineers to ensure that feedbacks are prevented in electronic circuits will, therefore, be relevant to a discussion of the role of feedbacks acting upon the climate object.
In the construction of electronic circuits, where one of the best-known instances of runaway feedback is the howling shriek when a microphone is placed too close to the loudspeaker to which it is connected, electronic engineers take considerable care to avoid positive feedback altogether, unless they wish to induce a deliberate instability or oscillation by compelling the loop gain to exceed unity, the singularity in Eq. (4), at which point the magnitude of the loop gain becomes undefined.
In electronic circuits for consumer goods, the values of components typically vary by up to 10% from specification owing to the vagaries of raw materials, manufacture, and assembly. Values may vary further over their lifetime from age and deterioration. Therefore engineers ensure long-term stability by designing in a negative feedback to ensure that vital circuit parameters stay close to the desired values.
Negative feedbacks were first posited by Harold S. Black in 1927 in New York, when he was looking for a way to cancel distortion in telephone relays. Roe (2009) writes:
“He describes a sudden flash of inspiration while on his commute into Manhattan on the Lackawanna Ferry. The original copy of the page of the New York Times on which he scribbled down the details of his brainwave a few days later still has pride of place at the Bell Labs Museum, where it is regarded with great reverence.”
One circuit parameter of great importance is the (closed) feedback loop gain inside any amplifier, which must be held at less than unity under all circumstances to avoid runaway positive feedback (g ≥ 1). The loop gain typically depends on the values of at least half a dozen components, and the actual value of each component may randomly vary. To ensure stability the design value of the feedback loop gain must be held one or two orders of magnitude below unity: g <0.1, or preferably <0.01.
Now consider the common view of the climate system as an engine for converting forcings to temperature changes – an object on which feedbacks act as in Fig. 1. The values of the parameters that determine the (closed) loop gain, as in an electronic circuit, are subject to vagaries. As the Earth evolves, continents drift, sometimes occupying polar or tropical positions, sometimes allowing important ocean currents to pass and sometimes impeding or diverting them; vegetation comes and goes, altering the reflective, radiative, and evaporative characteristics of the land and the properties of the coupled atmosphere-ocean interface; volcanoes occasionally fill the atmosphere with smoke, sulfur, or CO2; asteroids strike; orbital characteristics change slowly but radically in accordance with the Milankovich cycles; and atmospheric concentrations of the greenhouse species, vary greatly.
In the Neoproterozoic, 750 Ma BP, CO2 concentration (today <0.04%) was ~30%: otherwise the ocean’s magnesium ions could not have united with the abundance of calcium ions and with CO2 itself to precipitate the dolomitic rocks laid down in that era. Yet mile-high glaciers came and went twice at sea level at the equator.
As in the electronic circuit, so in the climate object, the values of numerous key components contributing to the loop gain change radically over time. Yet for at least 2 Ga the Earth appears never to have endured the runaway greenhouse warming that would have occurred if the loop gain had reached unity. Therefore, the loop gain in the climate object cannot be close to unity, for otherwise random mutation of the feedback-relevant parameters of vital climate components over time would surely by now have driven it to unity. It is near-certain, therefore, that the value of the climatic feedback loop gain g today must be very much closer to 0 than to 1.
A loop gain of 0.1, then, is in practice the upper bound for very-long-term climate stability. Yet the loop gain values implicit in the IPCC’s global-warming projections of 3.26[2, 4.5] K warming in response to a CO2 doubling are well above this maximum, at 0.64[0.42, 0.74] (Eq. 8). Values such as these are far too close to the steeply-rising segment of the climate-sensitivity curve (Fig. 3) to have allowed the climate to remain temperature-stable for hundreds of millions of years, as Zachos (2001) and Scotese (1999) have reported.
Figure 3. The climate-sensitivity curve at loop gains –1.0 ≤ g < +1.0. The narrow shaded zone at bottom left indicates that climate sensitivity is stable at 0.5-1.3 K per CO2 doubling for loop gains –1.0 ≤ g ≤ +0.1, equivalent to overall feedback gain factors 0.5 ≤ G ≤ 1.1. However, climate sensitivities on the IPCC’s interval [2.0, 4.5] K (shaded zone at right) imply loop gains on the interval (+0.4, +0.8), well above the maximum loop gain that could obtain in a long-term-stable object such as the climate. At a loop gain of unity, the singularity in the feedback-amplification equation (Eq. 4), runaway feedback would occur. If the loop gain in the climate object were >0.1, then at any time conditions sufficient to push the loop gain towards unity might occur, but (see Fig. 2) have not occurred in close to a billion years (author’s figure based on diagrams in Roe, 2009; Paltridge, 2009; and Lindzen, 2011).
Fig. 3 shows the climate-sensitivity curve for loop gains g on the interval [–1, 1). It is precisely because the IPCC’s implicit interval of feedback loop gains so closely approaches unity, which is the singularity in the feedback-amplification equation (Eq. 4), that attempts to determine climate sensitivity on the basis that feedbacks are strongly net-positive can generate very high (but physically unrealistic) climate sensitivities, such as the >10 K that Murphy et al. (2009) say they cannot rule out.
If, however, the loop gain in the climate object is no greater than the theoretical maximum value g = 0.1, then, by Eq. (4), the corresponding overall feedback gain factor G is 1.11, and, by Eq. (1), climate sensitivity in response to a CO2 doubling cannot much exceed 1.2 K. No surprise, then, that the dozen or more empirical methods of deriving climate sensitivity that I included in my commentary cohered at just 1 K. If that is indeed the answer to the climate sensitivity question, it is also a mortal blow to climate extremists worldwide – but good news for everyone else.
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