This is a guest essay by Mike Jonas, part 1 of 4
The aim of this article is to provide simple mathematical formulae that can be used to calculate the carbon dioxide (CO2) contribution to global temperature change, as represented in the computer climate models.
This article is the first in a series of four articles. Its purpose is to establish and verify the formulae, so unfortunately it is quite long and there’s a fair amount of maths in it. Parts 2 and 3 simply apply the formulae established in Part 1, and hopefully will be a lot easier to follow. Part 4 enters into further discussion. All workings and data are supplied in spreadsheets. In fact one aim is to allow users to play with the formulae in the spreadsheets.
Please note : In this article, all temperatures referred to are deg C anomalies unless otherwise stated.
Global Temperature Prediction
The climate model predictions of global temperature show on average a very slightly accelerating increase between +2 and +5 deg C by 2100:
We can be confident that all of this predicted temperature increase in the models is caused by CO2, because Skeptical Science (SkS) , following a discussion of CO2 radiative forcing, says :
Humans cause numerous other radiative forcings, both positive (e.g. other greenhouse gases) and negative (e.g. sulfate aerosols which block sunlight). Fortunately, the negative and positive forcings are roughly equal and cancel each other out, and the natural forcings over the past half century have also been approximately zero (Meehl 2004), so the radiative forcing from CO2 alone gives us a good estimate as to how much we expect to see the Earth’s surface temperature change.
So, if we can identify how much of the global temperature change over the years from 1850 to present was contributed by CO2, then we can deduce how much of the temperature change was not. ie,
T = Tc + Tn
T is temperature.
Tc is the cumulative net contribution to temperature from CO2. “CO2” refers to all CO2, there is no distinction between man-made and natural CO2.
Tn is the non-CO2 temperature contribution.
Obviously, all feedbacks to CO2 warming (changes which occur because the CO2 warmed) must be included in Tc.
The Denning Research Group  helpfully provide an emissions calculator, which shows CO2 levels and the estimated future temperature change that it causes under “Business as usual” (zero emission cuts) :
Cross-checking the future warming in this graph against Figure 1, the CO2 warming from 2020-2100 is just under 3.5 deg C compared with about 3.25 deg C model average in Figure 1. That seems close enough for reliable use here. But data going back to at least 1850 is still needed.
There is World Resources Institute (WRI) CO2 data from 1750 to present , and CO2 data measured at Mauna Loa from 1960 to present . Together with the Denning “Business as usual” CO2 predictions above, the CO2 concentration from 1750 to 2100 is as follows :
For dates covered by more than one series, the Mauna Loa measured data will be preferred, then the WRI data.
Because the “pre-industrial” CO2 is put at 280ppm , and the data points in the above graph before 1800 are all very close to 280ppm, a constant level of 280ppm will be assumed before 1800.
The only information still needed is the CO2-caused warming before about 1990.
A method for calculating the temperature contribution by CO2 is given by SkS in  :
dF = 5.35 ln(C/Co)
Where ‘dF’ is the radiative forcing in Watts per square meter, ‘C’ is the concentration of atmospheric CO2, and ‘Co’ is the reference CO2 concentration. Normally the value of Co is chosen at the pre-industrial concentration of 280 ppmv.
dT = λ*dF
Where ‘dT’ is the change in the Earth’s average surface temperature, ‘λ’ is the climate sensitivity, usually with units in Kelvin or degrees Celsius per Watts per square meter (°C/[W/m2]), and ‘dF’ is the radiative forcing.
So now to calculate the change in temperature, we just need to know the climate sensitivity. Studies have given a possible range of values of 2-4.5°C warming for a doubling of CO2 (IPCC 2007). Using these values it’s a simple task to put the climate sensitivity into the units we need, using the formulas above:
λ = dT/dF = dT/(5.35 * ln)= [2 to 4.5°C]/3.7 = 0.54 to 1.2°C/(W/m2)
Using this range of possible climate sensitivity values, we can plug λ into the formulas above and calculate the expected temperature change. The atmospheric CO2 concentration as of 2010 is about 390 ppmv. This gives us the value for ‘C’, and for ‘Co’ we’ll use the pre-industrial value of 280 ppmv.
dT = λ*dF = λ * 5.35 * ln(390/280) = 1.8 * λ
Plugging in our possible climate sensitivity values, this gives us an expected surface temperature change of about 1–2.2°C of global warming, with a most likely value of 1.4°C. However, this tells us the equilibrium temperature. In reality it takes a long time to heat up the oceans due to their thermal inertia. For this reason there is currently a planetary energy imbalance, and the surface has only warmed about 0.8°C. In other words, even if we were to immediately stop adding CO2 to the atmosphere, the planet would warm another ~0.6°C until it reached this new equilibrium state (confirmed by Hansen 2005). This is referred to as the ‘warming in the pipeline’.
Unfortunately, not enough exact parameters are given to allow the temperature contribution by CO2 to be calculated completely, because the effect of ocean thermal inertia has not been fully quantified. But it should be reasonable to derive the actual CO2 contribution by fitting the above formulae to the known data and to the climate model predictions.
The net radiation caused by CO2 is the downward infra-red radiation (IR) as described by SkS, less the upward IR from the CO2 warming already in the system (CWIS). This upward IR will be proportional to the fourth power of the absolute (deg K) value of CWIS . The net effect of CO2 on IR is therefore given by :
Rcy = 5.35 * ln(Cy/C0) – j * ((T0+Tcy-1)^4 – T0^4)
Rcy is the net downward IR from CO2 in year y.
Cy is the ppm CO2 concentration (C) in year y.
C0 is the pre-industrial CO2 concentration, ie. 280ppm.
j is a factor to be determined.
T0 is the base temperature (deg K) associated with C0.
Tcy is the cumulative CO2 contribution to temperature (Tc) at end year y, ie, CWIS.
For a doubling of CO2, in the absence of other natural factors, the equilibrium temperature increase using the SkS formula is 5.35 * ln(2) * λ where λ = 3.2/3.7 (assuming a mid-range equilibrium climate sensitivity (ECS) of 3.2).
At equilibrium, Rc = 0. For ECS = 3.2, j can therefore be determined from
0 = 5.35 * ln(2) – j * ((T0+3.2)^4 – T0^4)
j = (5.35 * ln(2)) / ((T0+3.2)^4 – T0^4)
Because “the natural forcings over the past half century have also been approximately zero” , the SST should be a reasonably good guide to CWIS. The global average SST 1981 to 2006 was 291.76 deg K . Subtracting the year 1993 Tc of 0.7 (from the Denning data ) gives T0=291 deg K. Hence j = (5.35 * ln(2)) / ((291+3.2)^4 – 291^4) = 1.16E-8 (ie. 1.16 * 10^-8, or 0.0000000116).
Using the formula
δTcy = k * Rcy
δTcy is the increase (deg C) in CWIS in year y.
k is the one-year impact on temperature per unit of net downward IR.
the value of k can be found which gives a future temperature increase matching that of the climate models, ie. 3.25 deg C from 2020 to 2100. A reasonability check is that the result should closely match but be slightly lower than the Denning warming calculation in Figure 2 (lower graph) (slightly lower because target is 3.25 deg C not 3.5) …..
….. it does, for k = 0.02611 (the graph for 3.5 deg C is also shown in ). [Note that the calculated warming is “anchored” at 1750 T=0, and that the shape is determined only by the formula so there is no guarantee that the 2020 and 2100 temperatures will be close to the Denning temperatures. ie, this is a genuine test.].
Note: At this rate, global temperature takes 52 years to get 80% of the way to equilibrium (as in “equilibrium climate sensitivity”), 75 years to reach 90%, 97 years to reach 95%, 148 years to reach 99%.
The above formula can therefore reliably be used for CO2’s contribution to global temperature since 1750.
1850 to 2100
The above formulae can now be applied to the period 1850 – 2100, to see how much has been and will be contributed to temperature by CO2.
For temperature data 1850 to present, Hadcrut4 global temperature  is used. For future temperatures, the formula warming (as in Figure 4) is used.
Applying the above formulae shows the contributions to temperature by CO2 and by other factors :
· the dominant contribution is from CO2,
· other factors contribute the inter-annual “wiggles” and virtually nothing else.
Note : The contribution to global temperature by CO2 is only man-made to the extent that the CO2 is man-made. As stated earlier, no distinction is made between man-made CO2 and natural CO2. Obviously, all pre-industrial CO2 was in fact natural. Similarly, for the non-CO2 contribution, no distinction is made between natural factors and non-CO2 man-made factors (such as land-clearing, for example), but the non-CO2 factors are thought to be predominantly natural. The feedbacks from the CO2 warming as claimed by the IPCC (eg. water vapour, clouds) are included in the CO2 contribution above.
The picture of global temperature and its drivers as presented by the IPCC and the computer models is one in which CO2 has been the dominant factor since the start of the industrial age, and natural factors have had minimal impact.
This picture is endorsed by organisations such as SkS and Denning. Using formulae derived from SkS, Denning and normal physics, this picture is now represented here using simple mathematical formulae that can be incorporated into a normal spreadsheet.
Anyone with access to a spreadsheet will be able to work with these formulae. It has been demonstrated above that the picture they paint is a reasonable representation of the CO2 calculations in the computer models.
The next articles in this series will look at applications of these formulae.
It is important to recognise that the formulae used here represent the internal workings of the climate models. There is no “climate denial” here, because the whole series of articles is based on the premise that the climate computer models are correct, using the mid-range ECS of 3.2.
See spreadsheet “Part1”  for the above calculations.
Mike Jonas (MA Maths Oxford UK) retired some years ago after nearly 40 years in I.T.
 Hadley Centre Hadcrut4 Global Temperature data http://www.metoffice.gov.uk/hadobs/hadcrut4/data/current/time_series/HadCRUT.126.96.36.199.annual_ns_avg.txt (Downloaded 20/5/2015)
 Climate model predictions from http://upload.wikimedia.org/wikipedia/commons/a/aa/Global_Warming_Predictions.png (Downloaded 20/5/2015) Note: Wikipedia is an unreliable source for contentious issues, but for factual information such as the output of computer models, and in the context for which it is being used here, it should be OK.
 CO2 data from 1750 to date is from World Resources Institute http://powerpoints.wri.org/climate/sld001.htm (Downloaded 20/5/2015. Digitised using xyExtract v5.1 (2011) by Wilton P Silva)
 Emissions calculator from Denning Research Group at Colorado State University http://biocycle.atmos.colostate.edu/shiny/emissions/ using no emissions cuts, ie, “Business as usual”. (Downloaded 20/5/2015. Digitised using xyExtract v5.1 (2011) by Wilton P Silva)
 Mauna Loa CO2 data from http://scrippsco2.ucsd.edu/data/flask_co2_and_isotopic/monthly_co2/monthly_mlf.csv (Downloaded 27/2/2012)
 Skeptical Science 3 Sep 2010 http://www.skepticalscience.com/Quantifying-the-human-contribution-to-global-warming.html (As accessed 20/5/2012).
 Spreadsheet “Part1” with all data and workings . Part1 (Excel .xlsx spreadsheet)
 Stefan-Boltzmann law. See http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/stefan.html
AR4 – (Fourth IPCC report)
AR5 – (Fifth IPCC report)
CO2 – Carbon Dioxide
CWIS – CO2 warming already in the system
ECS – Equilibrium Climate Sensitivity
IPCC – Intergovernmental Panel on Climate Change
IR – Infra-red (Radiation)
LIA – Little Ice Age
MWP – Medieval Warming Period
SKS – Skeptical Science (skepticalscience.com)
WRI – World Resources Institute