# Determination of the CO2 doubling global mean temperature from observations

Guest essay by Girma Orssengo, PhD

What is the expected global warming if the atmospheric CO2 concentration doubles from 400 to 800 ppm? The answer to this question is essential for our understanding of the earth’s climate. Unfortunately, the estimates for the CO2 doubling global mean temperature (GMT) vary by a factor of three from 1.5 oC to 4.5 oC. Data analysis in the literature shows that this very large uncertainty is due to the multi-decadal oscillation (MDO) of the GMT, and when this oscillation is removed, we find a time-invariant CO2 doubling GMT of 1.4 oC.

The aim of this article is to determine the time-invariant CO2 doubling GMT using published results in the scientific literature for the GMT and the observed atmospheric CO2 for the Mauna Loa data (Tans and Keeling, 2017).

The CO2 doubling GMT T2x is a parameter in a linear mathematical model that relates the logarithm of the annual atmospheric CO2 concentration ln(C) to the GMT T given by (Caldeira et al., 2003; Knutti and Hegerl, 2008; Wigley and Schlesinger, 1985)

Eq. 1

where Co is the atmospheric CO2 concentration corresponding to a reference GMT of T = 0. Note that since Eq. 1 could also be written as ln(C/Co) = (ln(2)/T2x)T, when C/Co = 2, this equation gives T = T2x. As a result, T2x in Eq. 1 represents the CO2 doubling GMT.

To estimate the CO2 doubling GMT directly from observations, the reference atmospheric carbon dioxide Co could be removed from Eq. 1 by differentiation of this equation with respect to year y, which gives

Eq. 2

Solving for the CO2 doubling GMT T2x in Eq. 2 gives

Eq. 3

Using the mathematical model given by Eq. 3, if for a given middle year of a trend period, the GMT trend dT/dy and the relative atmospheric carbon dioxide trend (dC/dy)/C are known, the CO2 doubling GMT T2x could be estimated directly form observations.

Under what condition could we determine the time-invariant CO2 doubling GMT from Eq. 3? We may answer this question by looking at the annual atmospheric CO2 data for Mauna Loa shown in Fig. 1, which suggests a monotonically increasing smoothed curve for the annual atmospheric CO2. From Eq. 2 and Fig. 1, to obtain a constant CO2 doubling GMT T2x, the GMT trend dT/dy must be proportional to the relative CO2 trend (dC/dy)/C at all times, which is only possible if the GMT T is also monotonically increasing like the annual atmospheric carbon dioxide C.

Fig. 1. Annual atmospheric CO2 for the Mauna Loa data (Tans, P. and Keeling, R., 2017) and an average atmospheric CO2 concentration of 343.32 ppm and a least squares average CO2 trend of 1.46 ppm/year for the trend period middle year of 1983.

Several studies (Delsole et al., 2011; Knudsen et al., 2011; Latif and Keenlyside, 2011; Schlesinger and Ramankutty, 1994; Swanson et al., 2009; Wu et al., 2011) have reported that the annual GMT data has multi-decadal oscillation (MDO) having 55 to 70 year period. As a result, before the annual GMT could be used in Eq. 3, its MDO must be removed. Wu et al. (2011) have reported the secular GMT trend obtained after removing the MDO from the annual GMT data as given in Table 1, which approximately corresponds to the annual atmospheric CO2 data period shown in Fig. 1 and 2. Note that the annual atmospheric CO2 data for Mauna Loa starts from 1959.

From the results of Wu et al. (2011) for the secular GMT trend dT/dy for a given trend period middle year given in Table 1 and the observed relative atmospheric carbon dioxide trend (dC/dy)/C for the same trend period middle year, the CO2 doubling GMT could be calculated using the mathematical model given by T2x = ln(2)(dT/dy)C/(dC/dy) (Eq. 3).

Table 1. Secular GMT trends obtained after removing the multi-decadal oscillation from the annual global mean temperature data (Wu et al. 2011).

 Trend period length (year) Trend Period Trend Period Middle Year Secular GMT Trend dT/dy (oC/year) 50 1958 to 2008 1983 0.0086 25 1983 to 2008 1995.5 0.0096

From Table 1, for the trend period middle year of 1983, the secular GMT trend dT/dy = 0.0086 oC/year. For the same trend period middle year of 1983, Fig. 1 shows an average atmospheric carbon dioxide concentration of C = 343.32 ppm and its average trend of dC/dy = 1.46 ppm/year. Substituting these values into Eq. 3 gives a CO2 doubling GMT of

oC Eq. 4

This result means that if the atmospheric CO2 were doubled from, say, 400 to 800 ppm, the secular GMT would increase by 1.4 oC.

To verify whether this CO2 doubling GMT of 1.4 oC determined for the trend period middle year of 1983 is time-invariant, we calculate its value for a different trend period middle year. From Table 1, for the trend period middle year of 1995.5, the secular GMT trend dT/dy = 0.0096 oC/year. For the same trend period middle year of 1995.5, Fig. 2 shows an average atmospheric carbon dioxide concentration of C = 363 ppm and its average trend of dC/dy = 1.67 ppm/year. Substituting these values into Eq. 3 gives a CO2 doubling GMT of

oC Eq. 5

The above results (Eq. 4 & 5) for two different time periods show that the time-invariant CO2 doubling GMT is 1.4 oC. This result is almost identical to the minimum possible CO2 doubling GMT of 1.5 oC reported in IPCC (2007): “The equilibrium climate sensitivity is a measure of the climate system response to sustained radiative forcing. It is not a projection but is defined as the global average surface warming following a doubling of carbon dioxide concentrations. It is likely to be in the range 2°C to 4.5°C with a best estimate of about 3°C, and is very unlikely to be less than 1.5°C.”

Regarding to the CO2 doubling GMT, in addition to its minimum possible value, IPCC (2007) had also reported: “It is likely to be in the range 2°C to 4.5°C with a best estimate of about 3°C”. How could we also determine these IPCC estimates using our mathematical model given by Eq. 3?

To determine IPCC’s CO2 doubling GMT estimates above using our empirical model, we use the GMT trends given in the same report (IPCC, 2007) : “Since IPCC’s first report in 1990, assessed projections have suggested global average temperature increases between about 0.15°C and 0.3°C per decade for 1990 to 2005. This can now be compared with observed values of about 0.2°C per decade, strengthening confidence in near-term projections.”

Note that these IPCC’s GMT trends of 0.02 and 0.03°C per year are much greater than the secular GMT trend of 0. 0096 oC per year reported by Wu et al (2011) given in Table 1.

The IPCC report quoted above suggests a central GMT trend of 0.02 oC/year. Replacing the secular GMT trend of dT/dy = 0.0096 oC/year in Eq. 5 from Wu et al. (2011) with IPCC’s central GMT trend of 0.02 oC/year gives

oC Eq. 6

Remarkably, this calculated value for the central IPCC GMT trend is identical to the central CO2 doubling GMT of 3 oC reported in IPCC (2007).

Fig. 2. An average atmospheric CO2 concentration of 363.00 ppm and a least squares average CO2 trend of 1.67 ppm/year for the trend period middle year of 1995.5 for the Mauna Loa data (Tans, P. and Keeling, R., 2017).

The IPCC report quoted above also suggests an upper GMT trend of 0.03 oC/year. Replacing IPCC’s central GMT trend of dT/dy = 0.02 oC/year in Eq. 6 with its upper GMT trend of 0.03 oC/year gives

oC Eq. 7

Remarkably, again, this calculated value for the upper IPCC trend is identical to the upper CO2 doubling GMT of 4.5 oC reported by IPCC quoted above.

Regarding the history for the range of values for the CO2 doubling GMT, Kerr has reported an interesting story (Kerr, 2004): “On the first day of deliberations, Manabe told the committee that his model warmed 2°C when CO2 was doubled. The next day Hansen said his model had recently gotten 4°C for a doubling. According to Manabe, Charney chose 0.5°C as a not-unreasonable margin of error, subtracted it from Manabe’s number, and added it to Hansen’s. Thus was born the 1.5°C-to-4.5°C range of likely climate sensitivity that has appeared in every greenhouse assessment since, including the three by the Intergovernmental Panel on Climate Change (IPCC). More than one researcher at the workshop called Charney’s now-enshrined range and its attached best estimate of 3°C so much hand waving.

In this article, we showed that Charney’s range are not “so much hand waving” because they could be determined using the mathematical model T2x = ln(2)(dT/dy)C/(dC/dy) (Eq. 3), the observed relative atmospheric CO2 trend (dC/dy)/C (Fig. 2), and IPCC’s GMT trends dT/dy.

In conclusion, we found a time-invariant CO2 doubling GMT of 1.4 oC (Eq. 4 & 5). We also showed that the higher CO2 doubling GMT values reported in IPCC (2007) are for secular GMT trends of 0.2 and 0.3 oC/decade that are inconsistent with the observed secular GMT trend of about 0.1 oC/decade (Delsole et al., 2011; Wu et al., 2011). Note that as the annual GMT has been reported to have a multi-decadal oscillation (MDO) of about 55 to 70 years for the last 8000 years (Knudsen et al., 2011), a linear trend of at least a 70-year period should be used to remove the contribution of the MDO to determine the secular GMT trend, which gives about 0.1 oC/decade for the latest 70-year period from 1946 to 2016.

From about 1960 to 1990 with a trough in the mid-1970s, the MDO was in its cool phase, and it has been in its warm phase since 1990 that is expected to continue until about 2020. In the early-2020s, the cool phase of the MDO is expected to start with its trough in mid-2030s. The empirical evidence for this drop in global mean surface temperature would be the recovery of arctic sea ice and cooling of the Northern Hemisphere for the period from about 2020 to 2050.

When we start to see a steady increase in arctic sea ice in the 2020s that continues until the 2050s, what would happen to the “Theory of Man Made Global Warming”?

References

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January 28, 2018 at 01:31PM