A dynamic analysis of global temperature data gives a value of ECS = 2.5C ± 0.5C . Values above 3.0C or below 2.0C are ruled out. This analysis is based on two assumptions: 1) That net climate forcing follows that used in CMIP5 (ref 1). 2) That climate equilibrium is reached with an e-folding time of 15 years (derived from GISS Model-II).
Equilibrium Climate Change or ECS is defined as the increase in global temperatures following a doubling of CO2, once the climate system has stabilised. Models can calculate ECS by running a step function for CO2 concentrations from say 280ppm to 560ppm and then plotting how the temperature responds with time. Each model gives a different value for ECS, and the spread in values represents in AR5 as an estimate of the uncertainty. I am going to use one of the simplest models, GISS Model II to investigate this lag effect of climate stabilisation which is mainly caused by the heat inertia response of the oceans to increased forcing.
After roughly 100 years the climate reaches a new stable state, and shows that GISS Model II gives a value for ECS of 4.4 C. The red curve is a fit to the temperature response curve which can be written in terms of temperature anomalies in the general form
In reality CO2 levels in the atmosphere have been slowly growing over the last 200 years by annual increments as recorded since 1950 by the Mauna Loa data. The direct radiative forcing from increased CO2 has been calculated by radiative transfer codes. My derivation of this formula is described here. A more precise parameterisation of that forcing is the well known formula
where is the initial CO2 concentration and C is the incremental value. A doubling of CO2 alone give a forcing of ~3.7 W/m2 which at equilibrium is balanced by a surface temperature rise of 1.1C by applying Stefan Boltzmann’s law.
Ignoring feedbacks and using results in ECS being ~ 1.1C. Higher values of ECS are due to net positive climate feedbacks, mainly from increased H2O. CMIP5 models give a large spread in predicted ECS values due to the different ways H2O and cloud feedbacks are handled. Can we measure ECS directly from the data ?
The problem with measuring ECS from the temperature data is that net forcing is increasing every year so we can never wait long enough for the climate to reach an equilibrium state. Given these constraints I adopt a different approach.
We treat the temperature record at any time as the response to the sum of previous discrete annual pulses of forcing. Each pulse causes a time dependent temperature response as shown in Figure 1. The resultant annual temperature for year n is then the integral of all previous responses up to that year.
Each pulse response is tracked through time and integrated to yield the overall instantaneous temperature at year N:
– Equation 1.
This procedure can then be repeated for various possible values of ECS and compared directly to the temperature data. Rather than using the CO2 forcing directly we use the ‘Total Anthropogenic’ AR5 forcing data as shown below, which turn out to be almost the same thing.
The equilibrium temperature response to an incremental forcing DS is , where f is calculated from each possible value of ECS by using:
where 3.7 is the direct forcing due to a doubling of CO2 (calculated from ) and f is the feedback parameter. This then allows to calculate the feedback parameter f corresponding to a particular value of ECS, and then use f to to calculate the impulse forcing response. The resultant values of f are as follows.
A perl script was written to integrate forward past temperature responses into a predicted annual temperature for various values of ECS by applying equation 1. The results are compared to the annual Hadcrut4.6 values.
It is instructive to look in more detail at the recent data as it then becomes obvious that high values and very low values of ECS are ruled out.
The best fit to the observed temperature distribution using this method is ECS = 2.5C. High values above 3.oC and very low values below 2.0C are ruled out. So my best estimate is
ECS = 2.5 ± 0.5C (95% probability)
The error is based on post 2000 temperature values. ECS=2.0 falls within just 12% of data point errors (0.05C) while ECS = 3.0 falls within 24%. This is to be compared with ECS=2.5C which falls within 84%. Both ECS=2.0 and ECS=3.0 are about 2 sigma from the mean average shown in black. By 2017 ECS=2.0 lies 0.15C below the mean and ECS=3.0 lies 0.26C above the mean. Therefore I estimate a 95% probability that ECS lies within this range.
If climate sensitivity is 2.5C then global temperatures can never rise more that 2.5C above pre-industrial levels so long as CO2 levels are kept below 560 ppm. This is a far more achievable goal than many activists are calling for since it requires only gradual reductions in CO2 emissions by 2100. This then gives us time to develop realistic alternatives, which I am convinced must have a strong nuclear base.
Forster et al. Evaluating adjusted forcing and model spread for historical and future scenarios in the CMIP5 generation of climate models, J. Geophys. Res.,118 1139-1150, 2013