This is a brief foray into the very controversial matter of the sensitivity of the climate to a change in radiative forcing. One experimental value and one mathematical value contribute to the discussion and are compared with the IPCC condsiderably higher value. A more recent empirical approach is also described.
On June 15, the eruption of
A value for the sensitivity of the atmosphere to a change in radiative forcing may be obtained by differentiating the Stefan-Boltzmann equation:
E = esT4
dE/dT = 4esT3
Thus, dT/dE = 1/(4esT3) = 0.19 K (W m-2)-1
Using 0.95 for the emissivity and 288.2 K as the surface temperature this gives:
dT/dE = 0.19 K (W m-2)-1
The value derived from the Stefan-Boltzmann equation is subject to the usual strictures; that it is an 'instantaneous' value and does not take into account any subsequent processes, neither positive nor negative feedbacks. The Pinatubo value has the advantage that it comes from a real event of world proportions, but suffers from the fact that the stratosphere was perturbed as well as the troposphere.
This much lower value for the sensitivity is more in keeping with other estimates in the literature and considerably lower than the value used by the IPCC modellers of 0.5 K (W m-2)-1. It is also consistent with estimates of the 20th century warming based on varying solar influences as well as that because of the extra CO2.
Solar influences are the subjects of a later part of this website presentation.
The Figure below illustrates the latitudinal distribution of incoming solar radiation and outgoing terrestrial radiation. From approximately 35o N to 35o S latitude there is a surplus of energy as incoming radiation exceeds outgoing. The more northerly and southerly regions indicate that there is more outgoing energy than incoming, yielding a net loss of energy from the Earth's surface. One might ask then why the middle to higher latitudes aren't getting colder through time as a result of the net loss, and the subtropical to equatorial regions getting constantly hotter due to the net gain. The reason is that the energy is redistributed by circulation of the atmosphere and oceans. Heat gained in the tropics is transported poleward by the global circulation of air and warm ocean currents to heat higher latitude regions. Cooler air from the higher latitudes and cold ocean currents push equatorward to cool the lower latitudes. This process of redistributing energy in the Earth system helps maintain a long-term energy balance. It is the basis of the subject known as climatology.
A study of satellite data gives a surprisingly simple empirical relationship between the intensity of the IR radiation emitted to space [E = TOA emission flux] and the surface temperature:
E = 203.3 + 2.09 × T
T is the surface temperature in degrees Celsius [°C].
The equation takes into account all possible forcings and feedbacks and is not dependent upon any theoretical considerations.
The mean value for the outgoing IR radiation intensity is 235 W m-2 and this is consistent with the mean surface temperature of 15.2°C given by the equation and that generally accepted from terrestrial measurements.
The temperature coefficient, 2.09 W m-2 K-1 gives a value for the sensitivity of the system to a forcing as:
ΔT/ΔF = 1/2.09 = 0.48 K (W m-2)-1
Applied to the forcing arising from a doubling of the pre-industrial concentration of CO2 of 285 ppmv to 570 ppmv estimated to be 3.7 W m-2 the sensitivity indicates a possible surface temperature increase of 0.48 × 3.7 = 1.8°C. That we are supposed to have had an increase of 0.8°C, there seems to be no danger in the possible forthcoming 1.0°C if indeed there is sufficient carbon to burn to produce the CO2 concentration of 570 ppmv.
The IPPC value for the sensitivity used by the GCMs is 0.5 K (W m-2)-1 before the application of the various feedbacks. The similarity between the two values is some indication that the feedbacks incorporated into the GCMs are overdone.
Mathematical form of the equation
There is an underlying logic in the form of the empirical equation in the expansion of the Stefan-Boltzmann law in the form:
E = σ(273.2 + T)4 = σ(273.2)4 + 4σ(273.2)3T + 6σ(273.2)2T2 + 4σ(273.2)T3 + σT4
The last three terms are relatively small and may be ignored; the equation reduces to the linear form:
E ˜ σ(273.2)4 + 4σ(273.2)3T = 315.9 + 4.63 T
This has the same form as the empirical equation and ignoring the three last terms in the fully expanded equation introduces an error of only 1.5%. The equation is numerically different from the empirical one discussed above and that is because of the greenhouse effect which ensures that the outgoing radiation intensity is smaller than that emitted at the surface.
Climate sensitivity as generally understood is the extent of global warming if the atmospheric concentration of CO2 were to double from 300 ppmv to 600 ppmv. The specific doubling is quoted because of the logarithmic relationship between absorption of radiation and the concentration of the absorbing substance. Other doublings would be expected to show different results.
The latest paper [J. Geophys. Res., 115, 20106, (2010)] concerning the attribution of greenhouse gas contributions to the greenhouse effect gives results shown in Table 1.
Table 1 Major contributions to the greenhouse effect
The authors [Schmidt, Ruedy, Miller & Lacis] comment that 'since the attribution of CO2 is closer to 20% than 2%, it might make more intuitive sense that changes in CO2 could be important for climate change'. This is a reference to Richard Lindzen's unsubstantiated comment in a review of the 1991 IPCC publication, Climate Change: The IPCC Scientific Assessment in which he claimed that '98% of the natural greenhouse effect' is due to water vapour and stratiform clouds, and 'less than 2%' for CO2. This comment has been proliferated by extreme sceptics of climate change projections even though there seems to be no justification for it except from Lindzen's statement.
They also comment that 'Nonetheless, climate sensitivity can only be properly assessed from examining changes in climate, not from the mean climatology alone'. Schmidt et al's comment refers to an important paper by Annan & Hargreaves, 2006, entitled 'Using multiple observationally based constraints to estimate climate sensitivity' Geophys. Res. Lett, 33, 25259. Annan & Hargreaves consider data from volcanic cooling, changes in radiative forcing around the last glacial maximum, and the 20th century warming. Their combined estimates of climate sensitivity are given in the form (1.7, 2.9, 4.9), these being the values (in °C) of the lower, median and higher limits. The probabilities of a value being below 1.7°C or higher than 4.9°C are only 5%.
Figure 1 shows the data for CO2 concentrations and temperature anomalies since 1850.
Figure 1 Global mean temperatures [HADCRUT3 data set] and CO2 concentrations
There is a clear connection between the two sets of data and the relevant correlation coefficient has a value of 0.93. This is often dismissed by some sceptics who quite rightly state that correlation does not prove causation, although there seems to be no similar opinions expressed when correlations between sunspots or cosmic rays and surface temperature are discussed. The statement does not prove that there is no causation. The physics and absorption spectroscopy determine that a higher concentration of CO2 causes the surface to have a lower effective emissivity and the consequence is a heating of both the surface and the troposphere. It is also clear that there are considerable variations in the temperature curve that do not connect with any similar changes in the CO2 curve. This means that there are other factors involved in the determination of the temperature rather than the single forcing agent, CO2.
The IPCC in their Third Assessment Report, Climate Change 2001, The Scientific Basis, page 358, give a formula for calculation of the change in atmospheric forcing from a change in CO2 concentration as:
ΔF = α ln(c2/c1) 
They give a value of 5.35 for the constant α, and c1 and c2 are the two concentrations of CO2, with c2 > c1. They also suggest that the relationship between ΔF and global mean surface temperature change is:
ΔTs = ΔF/2 
For the doubling of the CO2 concentration from 300 ppmv to 600 ppmv the increase in mean global surface temperature, these relationships give a value of 1.85°C, but the general circulation models include feedbacks such that the value is altered to a range from 1.5°C to 4.5°C, not too different from the Annan/Hargreaves (1.7, 2.9, 4.9) estimate, with 2.9°C being the most likely value.
To investigate further the CO2-surface temperature relationship it was assumed that surface temperature anomaly was proportional to the logarithm of the CO2 concentration:
ΔTs = α ln c + b 
The logarithmic dependence arises from the Beer-Lambert Law of Light Absorption. The HADCRUT3 data from 1850 to 2010 were used, together with the CO2 records from the Siple cores and Mauna Loa. The relationship between ΔTs and the CO2 concentration was found to be a reasonably good straight line with a correlation coefficient of 0.93. The equation of best fit is:
ΔTs = (2.66 ± 0.17) × ln c - (15.45 ± 0.95) 
The 'c' in equation  is the CO2 concentration in ppmv. The error ranges are for 95% confidence limits for the 160 data points. The intercept is consistent with the global mean temperature between 1961 and 1990 ─ the basis of the HADCRUT3 data for anomalies. Figure 2 shows the HADCRUT3 temperature data and a plot of equation  using the annual mean CO2 concentrations for each year.
Figure 2 Observed and calculated values for surface temperature anomalies
The fit is as good as with the actual CO2 data with a correlation coefficient of 0.93, and represents only the limited range of values. Again it is clear that other factors are responsible for the temperature changes and Figure 3 shows the deviations of the observed temperature curve from the calculated values.
Figure 3 Differences between the observed temperature data and the calculated values
The differences shown in Figure 3 between the actual temperature data and the calculated values amount to a root mean square proportion of 32% for climate-influencing factors other than CO2. When deconvoluted the data shown in Figure 3 could possibly give a variety of cycles that might be associated with climate-influencing mechanisms.
The differences shown in Figure 3 between the actual temperature data and the calculated values amount to a root mean square proportion of 32% for climate-influencing factors other than CO2. When deconvoluted, the data shown in Figure 4 could possibly give a variety of cycles or trends that might be associated with climate-influencing mechanisms. The most probable ‘natural’ cycle that is responsible for the differences shown in Figure 4 is that known as the Atlantic Multidecadal Oscillation (AMO). The AMO signal is derived from the patterns of sea surface temperature variability in the North Atlantic after any linear trend has been removed. The de-trending is intended to remove the influence of greenhouse gas-induced global warming from the analysis. However, it is possible that if the global warming signal is significantly non-linear in time (i.e. not just a linear increase), variations in the forced signal will leak into the AMO definition. Figure 5 shows a comparison between the difference data of Figure A4 and the AMO index. The AMO signal appears to have a cycle length of about 65 years consistent with cooling effects from 1875 to 1910 (35 years) and from 1943 to 1975 (32 years) and with warming effects from 1910 to 1943 (33 years) and from 1975 to 2004 (29 years). There are other major cycles that affect the global temperature record including the Pacific Decadal Oscillation and the El Niño-Southern Oscillation that could explain the remaining deviations shown in Figure 5.
Figure 5 Comparison of the data of Figure 4 (black) with the AMO index data (grey)
Figure 5 shows a fairly good fit between the two sets of data, r = 0.62, but there are differences indicating that the AMO influence is not the only one. Based upon root mean square values, the AMO influence accounts for 82.5% of the difference between the observed and calculated data which means that 17.5% has some other explanation. In this treatment the greenhouse warming effects of methane and nitrous oxide have been neglected. The greenhouse gases other than CO2 and water vapour account, with clouds included, account for 93% of the observed warming so the neglect of methane and nitrous oxide is not a significant error. Those gases have increased their concentration since 1850 but not in any cyclic manner.
If this interpretation is sound, it means that the best value for the climate sensitivity, rather than being 2.9°C for the doubling of the CO2 concentration, is that derived from the equation used - a value of (2.66 ± 0.17) × ln 2 = (1.84 ± 0.11)°C. The value is at the low end of the Annan/Hargreaves estimate and practically equal to the IPCC value before feedbacks are applied. For the 20th Century warming, when the CO2 concentration increased from 285 ppmv to 380 ppmv the equation gives an increase in global mean surface temperature of 0.77°C, fairly close to the 0.74°C of observed warming. The latter figure is derived from the difference in mean temperature anomalies for the years 1990 - 2000 and 1900 - 1910.
This empirical approach to the sensitivity problem eliminates the effects of feedbacks since they are all in operation over the time period of 160 years. For the warming expected when the CO2 concentration changes from c1 to c2 the following equation should be used:
ΔTs = (2.66 ± 0.17) × ln(c2/c1) 
Applied to the 570 ppmv concentration of CO2 which is double the pre-industrial value, the overall expected warming of (1.84 ± 0.11)°C seems not to pose any great threat to the planet and its inhabitants. From the present concentration of 390 ppmv to 570 ppmv there is a further warming of ~1°C expected and that will only occur if there is sufficient fossil fuel burned to cause the required increase of CO2 concentration.
The whole-planet approach to global mean temperature changes does not take into account the possible changes in the geographic regional distribution of temperature. The slightly small sensitivity derived in this paper could hide a possible doubling of the northern hemisphere anomaly to ~3.7°C, with the southern hemisphere unaffected due to the ocean 'thermostat'. Such a possible change does require to be taken seriously.
More discussion is on page 51.