Abstract
The Intergovernmental Panel on Climate Change (IPCC, 2007) concluded that anthropogenic CO2 emissions probably caused more than half of the “global warming” of the past 50 years and would cause further rapid warming. However, global mean surface temperature has not risen since 1998 and may have fallen since late 2001. The present analysis suggests that the failure of the IPCC’s models to predict this and many other climatic phenomena arises from defects in its evaluation of the three factors whose product is climate sensitivity:
Radiative forcing ΔF;
The no-feedbacks climate sensitivity parameter κ; and
The feedback multiplier ƒ.
Some reasons why the IPCC’s estimates may be excessive and unsafe are explained. More importantly, the conclusion is that, perhaps, there is no “climate crisis”, and that currently-fashionable efforts by governments to reduce anthropogenic CO2 emissions are pointless, may be ill-conceived, and could even be harmful.
The context
LOBALLY-AVERAGED land and sea surface absolute temperature TS has not risen since 1998 (Hadley Center; US National Climatic Data Center; University of Alabama at Huntsville; etc.). For almost seven years, TS may even have fallen (Figure 1). There may be no new peak until 2015 (Keenlysideet al., 2008).
The models heavily relied upon by the Intergovernmental Panel on Climate Change (IPCC) had not projected this multidecadal stasis in “global warming”; nor (until trained ex post facto) the fall in TS from 1940-1975; nor 50 years’ cooling in Antarctica (Doran et al., 2002) and the Arctic (Soon, 2005); nor the absence of ocean warming since 2003 (Lyman et al., 2006; Gouretski&Koltermann, 2007); nor the onset, duration, or intensity of the Madden-Julian intraseasonal oscillation, the Quasi-Biennial Oscillation in the tropical stratosphere, El Nino/La Nina oscillations, the Atlantic Multidecadal Oscillation, or the Pacific Decadal Oscillation that has recently transited from its warming to its cooling phase (oceanic oscillations which, on their own, may account for all of the observed warmings and coolings over the past half-century: Tsoniset al., 2007); nor the magnitude nor duration of multi-century events such as the Mediaeval Warm Period or the Little Ice Age; nor the cessation since 2000 of the previously-observed growth in atmospheric methane concentration (IPCC, 2007); nor the active 2004 hurricane season; nor the inactive subsequent seasons; nor the UK flooding of 2007 (the Met Office had forecast a summer of prolonged droughts only six weeks previously); nor the solar Grand Maximum of the past 70 years, during which the Sun was more active, for longer, than at almost any similar period in the past 11,400 years (Hathaway, 2004; Solankiet al., 2005); nor the consequent surface “global warming” on Mars, Jupiter, Neptune’s largest moon, and even distant Pluto; nor the eerily- continuing 2006 solar minimum; nor the consequent, precipitate decline of ~0.8 °C in TS from January 2007 to May 2008 that has canceled out almost all of the observed warming of the 20th century.
Figure 1
Mean global surface temperature anomalies (°C), 2001-2008
Since the phase-transition in mean global surface temperature late in 2001, a pronounced downtrend has set in. In the cold winter of 2007/8, record sea-ice extents were observed at both Poles. The January-to-January fall in temperature from 2007-2008 was the greatest since global records began in 1880. Data sources: Hadley Center monthly combined land and sea surface temperature anomalies; University of Alabama at Huntsville Microwave Sounding Unit monthly lower-troposphere anomalies; Linear regressions – – – – – – –
An early projection of the trend in TS in response to “global warming” was that of Hansen (1988), amplifying Hansen (1984) on quantification of climate sensitivity. In 1988, Hansen showed Congress a graph projecting rapid increases in TS to 2020 through “global warming” (Fig. 2):
Figure 2
Global temperature projections and outturns, 1988-2020
Hansen (1988) projected that global temperature would stabilize (A) if global carbon dioxide concentration were controlled from 1988 and static from 2000: otherwise temperature would rise rapidly (B-C). IPCC (1990) agreed (D). However, these projections proved well above the National Climate Data Center’s outturn (E-F), which, in contrast to the Hadley Center and UAH records (Fig. 1), show a modest rise in temperature from 1998-2007. If McKitrick (2007) (G,H) is correct that temperature since 1980 has risen at only half of the observed rate, outturn tracks Hansen’s CO2 stabilization case (A), although emissions have risen rapidly since 1988.
To what extent, then, has humankind warmed the world, and how much warmer will the world become if the current rate of increase in anthropogenic CO2 emissions continues? Estimating “climate sensitivity” – the magnitude of the change in TS after doubling CO2 concentration from the pre-industrial 278 parts per million to ~550 ppm – is the central question in the scientific debate about the climate. The official answer is given in IPCC (2007):
“It is very likely that anthropogenic greenhouse gas increases caused most of the observed increase in [TS] since the mid-20th century. … The equilibrium global average warming expected if carbon dioxide concentrations were to be sustained at 550 ppm is likely to be in the range 2-4.5 °C above pre-industrial values, with a best estimate of about 3 °C.”
Here as elsewhere the IPCC assigns a 90% confidence interval to “very likely”, rather than the customary 95% (two standard deviations). There is no good statistical basis for any such quantification, for the object to which it is applied is, in the formal sense, chaotic. The climate is “a complex, non-linear, chaotic object” that defies long-run prediction of its future states (IPCC, 2001), unless the initial state of its millions of variables is known to a precision that is in practice unattainable, as Lorenz (1963; and see Giorgi, 2005) concluded in the celebrated paper that founded chaos theory –
“Prediction of the sufficiently distant future is impossible by any method, unless the present conditions are known exactly. In view of the inevitable inaccuracy and incompleteness of weather observations, precise, very-long-range weather forecasting would seem to be non-existent.”.
The Summary for Policymakers in IPCC (2007) says –
“The CO2radiative forcing increased by 20% in the last 10 years (1995-2005).”
Natural or anthropogenic CO2 in the atmosphere induces a “radiative forcing” ΔF, defined by IPCC (2001: ch.6.1) asa change in net (down minus up) radiant-energy flux at the tropopause in response to a perturbation. Aggregate forcing is natural (pre-1750) plus anthropogenic-era (post-1750) forcing. At 1990, aggregate forcing from CO2 concentration was ~27 W m–2 (Kiehl&Trenberth, 1997). From 1995-2005, CO2 concentration rose 5%, from 360 to 378 W m–2, with a consequent increase in aggregate forcing (from Eqn. 3 below) of ~0.26 W m–2, or <1%. f =" ΔF2xλ," f =" (1">ΔF2xCO2≈ 5.35 ln 2 ≈ 3.708 W m–2. (3)
To ΔF2xCO2 is added the slightly net-negative sum of all other anthropogenic-era radiativeforcings, calculated from IPCC values (Table 1), to obtain total anthropogenic-era radiative forcing ΔF2x at CO2 doubling (Eqn. 3). Note that forcings occurring in the anthropogenic era may not be anthropogenic.
Table 1
Evaluation of ΔF2x from the IPCC’s anthropogenic-era forcings
Anthropogenic-era radiativeforcings from CO2, from long-lived (LLGHG) and short-lived (SLGHG) greenhouse gases are added to other forcings to yield total anthropogenic-era forcings ΔF2x, which are then reduced by a probability-density function. The column for 1750-2005 summarizes the values given in IPCC (2007). The column for forcings from 1750 to CO2 doubling proceeds differently, since IPCC (2007) does not publish projected values for individual forcings at CO2 doubling other than that for CO2 itself. However, IPCC (2001) projected that CO2forcings by 2050-2100, when CO2 doubling is expected, would represent 70-80% of all greenhouse-gas forcings. That projection is followed here, while non-greenhouse-gas forcings (which are strongly net-negative) are conservatively held constant. To preserve the focus on anthropogenic forcings, the IPCC’s minuscule estimate of the solar forcing during the anthropogenic era is omitted.
From the anthropogenic-era forcings summarized in Table 1, we obtain the first of the three factors –
ΔF2x≈ 3.405 W m–2. (4)
2. The base or “no-feedbacks” climate sensitivity parameter κ, where ΔTκ is the response of TS to radiativeforcings ignoring temperature feedbacks, ΔTλ is the response of TS to feedbacks as well as forcings, and b is the sum in W m–2 °K–1of all individual temperature feedbacks, is –
κ = ΔTκ / ΔF2x °K W–1 m2, by definition; (5)
= ΔTλ / (ΔF2x + bΔTλ) °K W–1 m2. (6)
In Eqn. (5), ΔTκ, estimated by Hansen (1984) and IPCC (2007) as 1.2-1.3 °K at CO2 doubling, is the change in surface temperature in response to a tropopausalforcing ΔF2x, ignoring any feedbacks.
ΔTκ is not directly measurable in the atmosphere because feedbacks as well as forcings are present. Instruments cannot distinguish between them. However, from Eqn. (2) we may substitute 1 / (1 – bκ) for f in Eqn. (1), rearranging terms to yield a useful second identity, Eqn. (6), expressing κin terms of ΔTλ, which is measurable, albeit with difficulty and subject to great uncertainty (McKitrick, 2007).
IPCC (2007) does not mention κ and, therefore, provides neither error-bars nor a “Level of Scientific Understanding” (the IPCC’s subjective measure of the extent to which enough is known about a variable to render it useful in quantifying climate sensitivity). However, its implicit value κ≈ 0.313 °K W–1 m2, shown in Eqn. 7, may be derived using Eqns. 9-10 below, showing it to be the reciprocal of the estimated “uniform-temperature” radiative cooling response –
“Under these simplifying assumptions the amplification [f] of the global warming from a feedback parameter [b] (in W m–2 °C–1) with no other feedbacks operating is 1 / (1 – [bκ–1]), where [–κ–1] is the ‘uniform temperature’ radiative cooling response (of value approximately –3.2 W m–2 °C–1; Bony et al., 2006). If n independent feedbacks operate, [b] is replaced by (λ1 + λ 2+ ... λ n).” (IPCC, 2007: ch.8, footnote).
Thus, κ≈ 3.2–1 ≈ 0.313°K W–1 m2. (7)
3. The feedback multiplier f is a unitless variable by which the base forcing is multiplied to take account of mutually-amplified temperature feedbacks. A “temperature feedback” is a change in TSthat occurs precisely because TShas already changed in response to a forcing or combination of forcings. An instance: as the atmosphere warms in response to a forcing, the carrying capacity of the space occupied by the atmosphere for water vapor increases near-exponentially in accordance with the Clausius-Clapeyron relation. Since water vapor is the most important greenhouse gas, the growth in its concentration caused by atmospheric warming exerts an additional forcing, causing temperature to rise further. This is the “water-vapor feedback”. Some 20 temperature feedbacks have been described, though none can be directly measured. Most have little impact on temperature. The value of each feedback, the interactions between feedbacks and forcings, and the interactions between feedbacks and other feedbacks, are subject to very large uncertainties.
Each feedback, having been triggered by a change in atmospheric temperature, itself causes a temperature change. Consequently, temperature feedbacks amplify one another. IPCC (2007: ch.8) defines f in terms of a form of the feedback-amplification function for electronic circuits given in Bode (1945), where b is the sum of all individual feedbacks before they are mutually amplified:
f = (1 – bκ)–1 (8)
= ΔTλ/ ΔTκ
Note the dependence of f not only upon the feedback-sum b but also upon κ –
ΔTλ =(ΔF + bΔTλ)κ
==> ΔTλ (1 – bκ) = ΔFκ
==> ΔTλ = ΔFκ(1 – bκ)–1
==> ΔTλ/ ΔF = λ = κ(1 – bκ)–1 = κf
==> f = (1 – bκ)–1 ≈ (1 – b /3.2)–1
==> κ ≈ 3.2–1 ≈ 0.313 °K W–1 m2. (9)
Equivalently, expressing the feedback loop as the sum of an infinite series,
ΔTλ = ΔFκ+ ΔFκ 2b + ΔFκ 2b2 + …
= ΔFκ(1 + κb + κb2 + …)
= ΔFκ(1 –κb)–1
= ΔFκf
==>λ = ΔTλ/ΔF = κf (10)
Figure 3
Bode (1945) feedback amplification schematic
A forcing dF is input by multiplication to the final or “with-feedbacks” climate sensitivity parameter λ = κf, yielding the output dT = dFλ = dFκf. To find λ = κf, the base or “no-feedbacks” climate sensitivity parameter κ is successively amplified round the feedback-loop by feedbacks summing to b.
For the first time, IPCC (2007) quantifies the key individual temperature feedbacks summing to b:
“In AOGCMs, the water vapor feedback constitutes by far the strongest feedback, with a multi-model mean and standard deviation … of 1.80 ± 0.18 W m–2K–1, followed by the negative lapse rate feedback (–0.84 ± 0.26 W m–2 K–1) and the surface albedo feedback (0.26 ± 0.08 W m–2 K–1). The cloud feedback mean is 0.69 W m–2 K–1with a very large inter-model spread of ±0.38 W m–2 K–1.” (Soden& Held, 2006).
To these we add the CO2 feedback, which IPCC (2007, ch.7) separately expresses not as W m–2°K–1 but as concentration increase per CO2 doubling: [25, 225] ppmv, central estimate q = 87 ppmv. Where p is concentration at first doubling, the proportionate increase in atmospheric CO2 concentration from the CO2 feedback is o = (p + q) / p = (556 + 87) / 556 ≈ 1.16. Then theCO2 feedback is –
λCO2= zln(o) / dTλ ≈ 5.35 ln(1.16) / 3.2 ≈ 0.25 W m–2 K–1. (11)
The CO2 feedback is added to the previously-itemized feedbacks to complete the feedback-sumb:
b = 1.8 – 0.84 + 0.26 + 0.69 + 0.25 ≈ 2.16 W m–2 ºK–1, (12)
so that, where κ= 0.313, the IPCC’s unstated central estimate of the value of the feedback factor f is at the lower end of the range f = 3-4 suggested in Hansen et al. (1984) –
f = (1 – bκ)–1≈(1 – 2.16 x 0.313)–1 ≈ 3.077. (13)
Final climate sensitivity ΔTλ,after taking account of temperature feedbacks as well as the forcings that triggered them,is simply the product of the three factors described in Eqn. (1), each of which we have briefly described above. Thus, at CO2 doubling, –
ΔTλ = ΔF2xκ f ≈ 3.405 x 0.313 x 3.077 ≈ 3.28 °K (14)
IPCC (2007) gives dTλon [2.0, 4.5] ºK at CO2 doubling, central estimate dTλ≈ 3.26 °K, demonstrating that the IPCC’s method has been faithfully replicated. There is a further checksum, –
ΔTκ = ΔTλ/ f = κ ΔF2x= 0.313 x 3.405 ≈ 1.1 °K, (15)
sufficiently close to the IPCC’s estimate ΔTκ ≈ 1.2 °K, based on Hansen (1984), who had estimated a range 1.2-1.3 °K based on his then estimate that the radiative forcing ΔF2xCO2 arising from a CO2 doubling would amount to 4.8 W m–2, whereas the IPCC’s current estimate is ΔF2xCO2 = 3.71 W m–2 (see Eqn. 2), requiring a commensurate reduction in ΔTκthat the IPCC has not made.
A final checksum is provided by Eqn. (5), giving a value identical to that of the IPCC at Eqn (7):
κ = ΔTλ / (ΔF2x + bΔTλ)
≈ 3.28 / (3.405 + 2.16 x 3.28)
≈ 0.313°K W–1 m2. (16)
Having outlined the IPCC’s methodology, we proceed to re-evaluate each of the three factors in dTλ. None of these three factors is directly mensurable. For this and other reasons, it is not possible to obtain climate sensitivity numerically using general-circulation models: for, as Akasofu (2008) has pointed out, climate sensitivity must be an input to any such model, not an output from it.
In attempting a re-evaluation of climate sensitivity, we shall face the large uncertainties inherent in the climate object, whose complexity, non-linearity, and chaoticity present formidable initial-value and boundary-value problems. We cannot measure total radiative forcing, with or without temperature feedbacks, because radiative and non-radiative atmospheric transfer processes combined with seasonal, latitudinal, and altitudinal variabilities defeat all attempts at reliable measurement. We cannot even measure changes in TS to within a factor of two (McKitrick, 2007).
Even satellite-based efforts at assessing total energy-flux imbalance for the whole Earth-troposphere system are uncertain. Worse, not one of the individual forcings or feedbacks whose magnitude is essential to an accurate evaluation of climate sensitivity is mensurable directly, because we cannot distinguish individual forcings or feedbacks one from another in the real atmosphere, we can only guess at the interactions between them, and we cannot even measure the relative contributions of all forcings and of all feedbacks to total radiative forcing. Therefore we shall adopt two approaches: theoretical demonstration (where possible); and empirical comparison of certain outputs from the models with observation to identify any significant inconsistencies.
Radiative forcing ΔF2x reconsidered
We take the second approach with ΔF2x. Since we cannot measure any individual forcing directly in the atmosphere, the models draw upon results of laboratory experiments in passing sunlight through chambers in which atmospheric constituents are artificially varied; such experiments are, however, of limited value when translated into the real atmosphere, where radiative transfers and non-radiative transports (convection and evaporation up, advection along, subsidence and precipitation down), as well as altitudinal and latitudinal asymmetries, greatly complicate the picture. Using these laboratory values, the models attempt to produce latitude-versus-altitude plots to display the characteristic signature of each type of forcing. The signature or fingerprint of anthropogenic greenhouse-gas forcing, as predicted by the models on which the IPCC relies, is distinct from that of any other forcing, in that the models project that the rate of change in temperature in the tropical mid-troposphere – the region some 6-10 km above the surface – will be twice or thrice the rate of change at the surface (Figure 4):
Figure 4
Temperature fingerprints of five forcings
Modeledzonal mean atmospheric temperature change (ºC per century, 1890-1999) in response to five distinct forcings (a-e), and to all five forcings combined (f). Altitude is in hPa (left scale) and km (right scale) vs. latitude (abscissa). Source: IPCC (2007).
The fingerprint of anthropogenic greenhouse-gas forcing is a distinctive “hot-spot” in the tropical mid-troposphere. Figure 4 shows altitude-vs.-latitude plots from four of the IPCC’s models:
Figure 5
Fingerprints of anthropogenic warming projected by four models
Zonal mean equilibrium temperature change (°C) at CO2 doubling (2x CO2 – control), as a function of latitude and pressure (hPa) for 4 general-circulation models. All show the projected fingerprint of anthropogenic greenhouse-gas warming: the tropical mid-troposphere “hot-spot” is projected to warm at twice or even thrice the surface rate. Source: Lee et al. (2007).
However, as Douglass et al. (2004) and Douglass et al. (2007) have demonstrated, the projected fingerprint of anthropogenic greenhouse-gas warming in the tropical mid-troposphere is not observed in reality. Figure 6 is a plot of observed tropospheric rates of temperature change from the Hadley Center for Forecasting. In the tropical mid-troposphere, at approximately 300 hPa pressure, the model-projected fingerprint of anthropogenic greenhouse warming is absent from this and all other observed records of temperature changes in the satellite and radiosonde eras:
Figure 6
The absent fingerprint of anthropogenic greenhouse warming
Altitude-vs.-latitude plot of observed relative warming rates in the satellite era. The greater rate of warming in the tropical mid-troposphere that is projected by general-circulation models is absent in this and all other observational datasets, whether satellite or radiosonde. Altitude units are hPa (left) and km (right). Source: Hadley Centre for Forecasting (HadAT, 2006).
None of the temperature datasets for the tropical surface and mid-troposphere shows the strong differential warming rate predicted by the IPCC’s models. Thorne et al. (2007) suggested that the absence of the mid-tropospheric warming might be attributable to uncertainties in the observed record: however, Douglass et al. (2007) responded with a detailed statistical analysis demonstrating that the absence of the projected degree of warming is significant in all observational datasets.
Allen et al. (2008) used upper-atmosphere wind speeds as a proxy for temperature and concluded that the projected greater rate of warming at altitude in the tropics is occurring in reality. However, satellite records, such as the RSS temperature trends at varying altitudes, agree with the radiosondes that the warming differential is not occurring: they show that not only absolute temperatures but also warming rates decline with altitude.
There are two principal reasons why the models appear to be misrepresenting the tropical atmosphere so starkly. First, the concentration of water vapor in the tropical lower troposphere is already so great that there is little scope for additional greenhouse-gas forcing. Secondly, though the models assume that the concentration of water vapor will increase in the tropical mid-troposphere as the space occupied by the atmosphere warms, advection transports much of the additional water vapor poleward from the tropics at that altitude.
Since the great majority of the incoming solar radiation incident upon the Earth strikes the tropics, any reduction in tropical radiative forcing has a disproportionate effect on mean global forcings. On the basis of Lindzen (2007), the anthropogenic-ear radiative forcing as established in Eqn. (3) are divided by 3 to take account of the observed failure of the tropical mid-troposphere to warm as projected by the models –
ΔF2x≈ 3.405 / 3 ≈ 1.135 W m–2. (17)
The “no-feedbacks” climate sensitivity parameter κ reconsidered
The base climate sensitivity parameter κis the most influential of the three factors of ΔTλ: for the final or “with-feedbacks” climate sensitivity parameter λ is the product of κand the feedback factor f, which is itselfdependent not only on the sum b of all climate-relevant temperature feedbacks but also on κ.Yet κ has received limited attention in the literature. In IPCC (2001, 2007) it is not mentioned. However, its value may be deduced from hints in the IPCC’s reports. IPCC (2001, ch. 6.1) says:
“The climate sensitivity parameter (global mean surface temperature response ΔTS to the radiative forcing ΔF) is defined as ΔTS / ΔF = λ {6.1} (Dickinson, 1982; WMO, 1986; Cesset al., 1993). Equation {6.1} is defined for the transition of the surface-troposphere system from one equilibrium state to another in response to an externally imposed radiative perturbation. In the one-dimensional radiative-convective models, wherein the concept was first initiated, λis a nearly invariant parameter (typically, about 0.5 °K W−1 m2; Ramanathanet al., 1985) for a variety of radiativeforcings, thus introducing the notion of a possible universality of the relationship between forcing and response.”
Since λ= κf = κ(1 – bκ)–1 (Eqns. 1, 2), where λ = 0.5 °K W–1 m2 and b ≈ 2.16 W m–2 °K–1 (Eqn. 12), it is simple to calculate that, in 2001, one of the IPCC’s values for f was 2.08. Thus the value f = 3.077 in IPCC (2007) represents a near-50% increase in the value of f in only five years. Where f = 2.08, κ = λ / f ≈ 0.5 / 2.08 ≈ 0.24 °K W–1 m2, again substantially lower than the value implicit in IPCC (2007). Some theory will, therefore, be needed.
The fundamental equation of radiative transfer at the emitting surface of an astronomical body, relating changes in radiant-energy flux to changes in temperature, is the Stefan-Boltzmann equation –
F = ε σ T4 W m–2, (18)
whereF is radiant-energy flux at the emitting surface; εis emissivity, set at 1 for a blackbody that absorbs and emits all irradiance reaching its emitting surface (by Kirchhoff’s law of radiative transfer, absorption and emission are equal and simultaneous), 0 for a whitebody that reflects all irradiance, and (0, 1) for a graybody that partly absorbs/emits and partly reflects; and σ ≈ 5.67 x 10–8 is the Stefan-Boltzmann constant.
Differentiating Eqn. (18) gives –
κ=dT / dF= (dF / dT)–1=(4 ε σ T3)–1 °K W–1 m2. (19)
Outgoing radiation from the Earth’s surface is chiefly in the near-infrared. Its peak wavelength λmax is determined solely by the temperature of the emitting surface in accordance with Wien’s Displacement Law, shown in its simplest form in Eqn. (20):
λmax= 2897 / TS = 2897 / 288 ≈ 10 μm. (20)
Since the Earth/troposphere system is a blackbody with respect to the infrared radiation that Eqn. (20) shows we are chiefly concerned with, we will not introduce any significant error if ε = 1, giving the blackbody form of Eqn. (19) –
κ= dT / dF = (4σ T3)–1 °K W–1 m2. (21)
At the Earth’s surface, TS≈ 288 °K, so that κS≈ 0.185 °K W–1 m2. At the characteristic-emission level, ZC, the variable altitude at which incoming and outgoing radiative fluxes balance, TC≈ 254 °K, so that κC ≈ 0.269 °K W–1 m2. The value κC ≈ 0.24, derived from the typical final-sensitivity value λ = 0.5 given in IPCC (2001), falls between the surface and characteristic-emission values for κ.
However, the IPCC, in its evaluation of κ, does not follow the rule that in the Stefan-Boltzmann equation the temperature and radiant-energy flux must be taken at the same level of the atmosphere. The IPCC’s value for κ is dependent upon temperature at the surface and radiant-energy flux at the tropopause, so that its implicit value κ≈ 0.313 °K W–1 m2 is considerably higher than either κS or κC.
IPCC (2007) cites Hansen et al. (1984), who say –
“Our three-dimensional global climate model yields a warming of ~4 ºC for … doubled CO2. This indicates a net feedback factor f= 3-4, because [the forcing at CO2 doubling] would cause the earth's surface temperature to warm 1.2-1.3 ºC to restore radiative balance with space, if other factors remained unchanged.”
Hansen says dF2x is equivalent to a 2% increase in incoming total solar irradiance (TSI). Top-of-atmosphere TSI S ≈ 1368 W m2, albedoα = 0.31, and Earth’s radius is r. Then, at the characteristic emission level ZC,
FC= S(1 – α)(πr2 / 4πr2) ≈ 1368 x 0.69 x (1/4) ≈ 236 W m–2. (22)
Thus a 2% increase in FCis equivalent to 4.72 W m–2, rounded up by Hansen to 4.8 W m–2, implying that κ ≈ 1.25 / 4.8 ≈ 0.260 °K W–1 m2. However, Hansen, in his Eqn. {14}, prefers 0.29 W m–2.
Bony et al. (2006), also cited by IPCC (2007), do not state a value for κ. However, they say –
“The Planck feedback parameter [equivalent to κ–1] is negative (an increase in temperature enhances the long-wave emission to space and thus reduces R [the Earth’s radiation budget]), and its typical value for the earth’s atmosphere, estimated from GCM calculations (Colman 2003; Soden and Held 2006), is ~3.2 W m2ºK–1 (a value of ~3.8 W m2ºK–1 is obtained by defining [κ–1]simply as 4σT3, by equating the global mean outgoing long-wave radiation to σT4 and by assuming an emission temperature of 255 ºK).”
Bony takes TC≈ 255 °Kand FC≈ 235 W m–2 at ZC as the theoretical basis for the stated prima facie value κ–1≈TC / 4FC≈3.8 W m2ºK–1, so that κ≈ 0.263 ºK W–1 m2, in very close agreement with Hansen. However, Bony cites two further papers, Colman (2003) and Soden& Held (2006), as justification for the value κ–1≈ 3.2 W m2ºK–1, so that κ ≈ 0.313 ºK W–1 m2.
Colman (2003) does not state a value for κ, but cites Hansen et al. (1984), rounding up the value κ ≈ 0.260 °K W–1 m2 to 0.3 °K W–1 m2 –
“The method used assumes a surface temperature increase of 1.2 °K with only the CO2 forcing and the ‘surface temperature’ feedback operating (value originally taken from Hansen et al. 1984).”
Soden& Held (2006) likewise do not declare a value for κ. However, we may deduce their implicit central estimate κ≈ 1 / 4 ≈ 0.250 °K W–1 m2 from the following passage –
“The increase in opacity due to a doubling of CO2 causes [the characteristic emission level ZC] to rise by ~150 meters. This results in a reduction in the effective temperature of the emission across the tropopause by ~(6.5K/km)(150 m) ≈ 1 K, which converts to 4 W m–2 using the Stefan-Boltzmann law.”
Thus the IPCC cites only two papers that cite two others in turn. None of these papers provides any theoretical or empirical justification for a value as high as the κ ≈ 0.313 °K W–1 m2 chosen by the IPCC.
Kiehl (1992) gives the following method, where FCis total flux at ZC:
κS= TS/ (4FC)≈ 288 / (4 x 236) ≈ 0.305 °K W–1 m2. (23)
Hartmann (1994) echoes Kiehl’s method, generalizing it to any level J of an n-level troposphere thus:
κJ = TJ/ (4FC)
= TJ/ [S(1 – α)]
≈ TJ/ [1368(1 – 0.31)] ≈ TJ/ 944 °K W–1 m2. (24)
Table 2 summarizes the values of κevident in the cited literature, with their derivations, minorespriores. The greatest value, chosen in IPCC (2007), is 30% above the least, chosen in IPCC (2001). However, because the feedback factor f depends not only upon the feedback-sum b≈ 2.16 W m–2°K–1but also upon κ, the 30% increase in κ nearly doubles final climate sensitivity:
Table 2
Values of the “no-feedbacks” climate sensitivity parameter κ
The range of values for κ in the IPCC’s assessment reports and in the papers which it cites is substantial. The value of κ implicit in IPCC (2007) is some 30% above that which is implicit in IPCC (2001): consequently, the value of the climate-sensitivity parameter λ is almost doubled. Though it is usual to assume a constant temperature lapse-rate, and hence to use the value of κ that obtains at the characteristic-emission level, where inbound and outbound radiative fluxes balance by definition, the theIPCC’s current value for κ assumes that the lapse-rate increases as temperature rises. Also, the IPCC does not sufficiently allow for latitudinal asymmetry in distribution of the values of κ.
The value of κ cannot be deduced by observation, because temperature feedbacks are present and cannot be separately measured. However, it is possible to calculate κ using Eqn. (6), provided that the temperature change ΔTλ, radiativeforcingsΔF2x, and feedback-sum b over a given period are known. The years 1980 and 2005 will be compared, giving a spread of a quarter of a century. We take the feedback-sum b = 2.16 W m–2°K–1 and begin by establishing values for ΔF and ΔT:
CO2 concentration: 338.67 ppmv 378.77 ppmv ΔF = 5.35 ln (378.77/338.67) = 0.560 W m–2
Anomaly in TS: 0.144 °K 0.557 °K ΔT = 0.412 °K (NCDC)
Anomaly halved: ΔT = 0.206°K (McKitrick) (25)
CO2 concentrations are the annual means from 100 stations (Keeling & Whorf, 2004, updated). TS values are NCDC annual anomalies, as five-year means centered on 1980 and 2005 respectively. Now, depending on whether the NCDC or implicit McKitrick value is correct, κmay be directly evaluated:
NCDC: κ= ΔT/ (ΔF + bΔT) = 0.412 / (0.560 + 2.16 x 0.412) = 0.284 °K W–1 m2
McKitrick: κ= ΔT/ (ΔF + bΔT) = 0.206 / (0.599 + 2.16 x 0.206) = 0.197 °K W–1 m2
Mean: κ = (0.284 + 0.197) / 2 = 0.241 °K W–1 m2 (26)
We assume that Chylek (2008) is right to find transient and equilibrium climate sensitivity near-identical; that allof the warming from 1980-2005 was anthropogenic; that the IPCC’s values for forcings and feedbacks are correct; and, in line 2, that McKitrick is right that the insufficiently-corrected heat-island effect of rapid urbanization since 1980 has artificially doubled the true rate of temperature increase in the major global datasets.
With these assumptions, κ is shown to be less, and perhaps considerably less, than the value implicit in IPCC (2007). The method of finding κ shown in Eqn. (24), which yields a value very close to that of IPCC (2007), is such that progressively smaller forcing increments would deliver progressively larger temperature increases at all levels of the atmosphere, contrary to the laws of thermodynamics and to the Stefan-Boltzmann radiative-transfer equation (Eqn. 18), which mandate the opposite.
It is accordingly necessary to select a value for κthat falls well below the IPCC’s value. Dr. David Evans (personal communication, 2007) has calculated that the characteristic-emission-level value of κ should be diminished by ~10% to allow for the non-uniform latitudinal distribution of incoming solar radiation, giving a value near-identical to that in Eqn. (26), and to that implicit in IPCC (2001), thus –
κ = 0.9TC / [S(1 – α)]
≈0.9 x 254 / [1368(1 – 0.31)]≈ 0.242 °K W–1 m2 (27)
The feedback factor f reconsidered
The feedback factor f accounts for two-thirds of all radiative forcing in IPCC (2007); yet it is not expressly quantified, and no “Level Of Scientific Understanding” is assigned either to f or to the two variables b and κ upon which it is dependent.
Several further difficulties are apparent. Not the least is that, if the upper estimates of each of the climate-relevant feedbacks listed in IPCC (2007) are summed, an instability arises. The maxima are –
Water vapor feedback 1.98 W m–2 K–1
Lapse rate feedback –0.58 W m–2 K–1
Surface albedo feedback 0.34 W m–2 K–1
Cloud albedo feedback 1.07 W m–2 K–1
CO2 feedback 0.57 W m–2 K–1
Total feedbacks b 3.38 W m–2 K–1 (28)
Since the equation [f = (1 – bκ)–1] → ∞ as b → [κ–1 = 3.2 W m–2 K–1], the feedback-sum b cannot exceed 3.2 W m–2 K–1 without inducing a runaway greenhouse effect. Since no such effect has been observed or inferred in more than half a billion years of climate, since the concentration of CO2 in the Cambrian atmosphere approached 20 times today’s concentration, with an inferred mean global surface temperature no more than 7 °K higher than today’s (Figure 7), and since a feedback-induced runaway greenhouse effect would occur even in today’s climate where b >= 3.2 W m–2 K–1 but has not occurred, the IPCC’s high-end estimates of the magnitude of individual temperature feedbacks are very likely to be excessive, implying that its central estimates are also likely to be excessive.
Figure 7
Fluctuating CO2 but stable temperature for 600m years
Millions of years before present
Throughout the past 600 million years, almost one-seventh of the age of the Earth, the mode of global surface temperatures was ~22 °C, even when carbon dioxide concentration peaked at 7000 ppmv, almost 20 times today’s near-record-low concentration. If so, then the instability inherent in the IPCC’s high-end values for the principal temperature feedbacks has not occurred in reality, implying that the high-end estimates, and by implication the central estimates, for the magnitude of individual temperature feedbacks may be substantial exaggerations. Source: Temperature reconstruction by C.R. Scotese; CO2 reconstruction after R.A. Berner; see also IPCC (2007).
Since absence of correlation necessarily implies absence of causation, Figure 7 confirms what the recent temperature record implies: the causative link between changes in CO2 concentration and changes in temperature cannot be as strong as the IPCC has suggested. The implications for climate sensitivity are self-evident. Figure 7 indicates that in the Cambrian era, when CO2 concentration was ~25 times that which prevailed in the IPCC’s reference year of 1750, the temperature was some 8.5 °C higher than it was in 1750. Yet the IPCC’s current central estimate is that a mere doubling of CO2 concentration compared with 1750 would increase temperature by almost 40% of the increase that is thought to have arisen in geological times from a 20-fold increase in CO2 concentration (IPCC, 2007).
How could such overstatements of individual feedbacks have arisen? Not only is it impossible to obtain empirical confirmation of the value of any feedback by direct measurement; it is questionable whether the feedback equation presented in Bode (1945) is appropriate to the climate. That equation was intended to model feedbacks in linear electronic circuits: yet many temperature feedbacks – the water vapor and CO2 feedbacks, for instance – are non-linear. Feedbacks, of course, induce non-linearity in linear objects: nevertheless, the Bode equation is valid only for objects whose initial state is linear. The climate is not a linear object: nor are most of the climate-relevant temperature feedbacks linear. The water-vapor feedback is an interesting instance of the non-linearity of temperature feedbacks. The increase in water-vapor concentration as the space occupied by the atmosphere warms is near-exponential; but the forcing effect of the additional water vapor is logarithmic. The IPCC’s use of the Bode equation, even as a simplifying assumption, is accordingly questionable.
IPCC (2001: ch.7) devoted an entire chapter to feedbacks, but without assigning values to each feedback that was mentioned. Nor did the IPCC assign a “Level of Scientific Understanding” to each feedback, as it had to each forcing. In IPCC (2007), the principal climate-relevant feedbacks are quantified for the first time, but, again, no Level of Scientific Understanding” is assigned to them, even though they account for more than twice as much forcing as the greenhouse-gas and other anthropogenic-era forcings to which “Levels of Scientific Understanding” are assigned.
Now that the IPCC has published its estimates of the forcing effects of individual feedbacks for the first time, numerous papers challenging its chosen values have appeared in the peer-reviewed literature. Notable among these are Wentz et al. (2007), who suggest that the IPCC has failed to allow for two-thirds of the cooling effect of evaporation in its evaluation of the water vapor-feedback; and Spencer (2007),who points out that the cloud-albedo feedback, regarded by the IPCC as second in magnitude only to the water-vapor feedback, should in fact be negative rather than strongly positive.
It is, therefore, prudent and conservative to restore the values κ≈ 0.24 and f ≈ 2.08 that are derivable from IPCC (2001), adjusting the values a little to maintain consistency with Eqn. (27). Accordingly, our revised central estimate of the feedback multiplier f is –
f = (1 – bκ)–1≈(1 – 2.16 x 0.242)–1≈ 2.095 (29)
Final climate sensitivity
Substituting in Eqn. (1) the revised values derived for the three factors in ΔTλ, our re-evaluated central estimate of climate sensitivity is their product –
ΔTλ= ΔF2x κ f ≈ 1.135 x 0.242 x 2.095 ≈ 0.58 °K (30)
Theoretically, empirically, and in the literature that we have extensively cited, each of the values we have chosen as our central estimate is arguably more justifiable – and is certainly no less justifiable – than the substantially higher value selected by the IPCC. Accordingly, it is very likely that in response to a doubling of pre-industrial carbon dioxide concentration TS will rise not by the 3.26 °K suggested by the IPCC, but by <1>
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