The strange case of TCR and ECS

In this post we consider the strange coincidence that the net forcing used by CMIP5 models is essentially the same as CO2 forcing alone. This allows us to derive a value of TCR(Transient Climate Response) just from observational data.  Measuring ECS(EquilibriumClimate Sensitivity) however requires modeling information. We use the average CMIP5 forcing and a model derived “hysteresis” function in order to determine ECS from temperature data. The resulting energy imbalance calculated using these values of ECS and TCR is found to be the same as that derived by other methods.

The net climate forcing is mainly due to changes in anthropogenic GHGs and in aerosols. Something like 20-40% of aerosols are of anthropogenic origin. Aerosols have 3 main effects:

  1. They scatter incoming solar radiation cooling the earth.
  2. They (e.g. black carbon) absorb both incoming solar radiation and surface IR radiation
  3. They help seed clouds formation – net cooling effect.

Energy imbalance Q = F -\lambda\Delta{T}  where \lambda is the aerosol feedback. Models trade off aerosols against Climate Sensitivity to match observed temperatures. Aerosols are essentially the tuning parameter that match GCMs in hindcasts to previous surface temperatures. AR5 admits that they have  “low confidence” in the aerosol-cloud interaction, and the estimated uncertainties are that the net effect of aerosols could even be zero. However, if aerosol forcing is reduced then model sensitivities would be far too high.  I argued in the previous post that climate sensitivity should be defined as the measured temperature change for a measured doubling of CO2. Instead IPCC has defined it as the simulated change in temperature due to CO2 forcing alone, excluding other GHGs and aerosols. The amazing fact however is  that it doesn’t matter! In order for the models to agree with observed temperature rises since 1850 there is a near perfect cancelation between other GHGs and aerosols! Figure 1 shows net CMIP5 forcings compared to a CO2 only forcing.

Fig2: Comparison of a pure CO2 GHG forcing and the CMIP5 avergaed forcings used to hindcast past temperature dependency since 1850.

Fig1: Comparison of a pure CO2 GHG forcing and the CMIP5 averaged forcings used to hindcast past temperature dependency since 1850.

So therefore it doesn’t really matter whether we use GCM models to derive a value for TCR or simply fit the temperature data instead. Let’s do that and derive a value for TCR using

DT = \lambda DS and
DS = 5.3ln(C02/290) where 290 is the CO2 value in 1850.
so DT = \lambda *5.3ln(CO2/290)

For CO2 I take the Mauna Loa data smoothly interpolated back to a value of 290 in 1850. We then fit the temperature data to a ln(CO2/290) term. The result is shown below


A fit of the temperature anomaly data to lambda*5.3ln(CO2/290)

This gives a climate response value \lambda = 0.47 degC/Wm2     therefore

TCR = \lambda *3.7 = 1.7C

A  fit to the temperature data which includes a 60 year natural oscillation, possibly linked to AMO (see recent post by Bob Tisdale) , gives a slightly different result.

Fig 3: Overall fit to 164 years of global temperature data (HADCRUT4)

Fig 3: Overall fit to 164 years of global temperature data (HADCRUT4)

A part of the rapid warming from 1970 to 2000 can be seen as potentially due to the upturn in this oscillation. The CO2 component now has a lower climate response with a \lambda  = 0.41 degC/Wm2  and

TCR = \lambda *3.7 = 1.5C

Figure 1 shows that the effective average forcing from all CMIP5 models has essentially been the same as that from CO2 forcing alone. This means we can derive TCR as defined by IPCC through this coincidence. This remains true now even if the ratio of aerosols to other GHGs were to change in the future. These two  analysis essentially measure a value:

TCR = 1.6 ± 0.2 C   where the error is an estimate of the spread in fits.

Equilibrium Climate Response (ECS)

“ECS is defined as the change in global mean temperature, T2x, that results when the climate system, or a climate model, attains a new equilibrium with the forcing change F2x resulting from a doubling of the atmospheric CO2 concentration.” It is the temperature reached after the earth has restored energy balance following a doubling of CO2.  The observed global temperatures since 1850 are instantaneous measurements while the earth is “warming”. The cause of the delay is because the oceans have a huge thermal capacity. One way to estimate ECS is to “measure” the change in heat content of the oceans \Delta{Q} . Then

ECS = \frac{F_{2x}\Delta{T}}{\Delta{F}-\Delta{Q}}

However there is another way to do it by “measuring” instead the response of the earth to a sudden increase in forcing. I used an old GISS model to measure that inertia from a model  run which instantaneously doubles CO2. The temperature response is shown in figure 4 where the red curve is a fit I made to a (1-e^{\frac{t}{\tau}}) term.


Fig 1: Response temperature curve from a pulse doubling of CO2 in 1958 and fit described i the text

Fig 4: Response temperature curve from a pulse doubling of CO2 in 1958 and fit described i the text

T(t) = T_0 + \Delta{T_0}(1-e^\frac{-t}{15})

This provides a method to derive ECS from the temperature data once the net forcing is known.

where \Delta{T}_{0}  is the equilibrium temperature response to a change in forcing \Delta{S}  .

To calculate the CO2 forcing  I take a yearly increment of

\Delta{S} = 5.3 log (\frac{C}{C_0})  ,     where  C and C0 are the values before and after the yearly pulse. All values are calculated from seasonally averaged Mauna Loa data smoothed back to an initial concentration of 280ppm in 1750.

Each pulse is tracked through time and integrated into the overall transient temperature change using:

\Delta{T}(t) = \sum_{k=1}^N (\Delta{T}_{0}(1 - e^\frac{-t}{15}))

\Delta{T}_{0} was calculated based on different values of ECS.  The results are compared to the observed HADCRUT4 anomaly measurements in Figure 4. The publication of AR5 report allows us to update CMIP5 forcings up to 2013 based on this graph.



The data was extended from 2005 to increase forcing and agree with the data – black curve from AR5. The final net forcing in 2013 is 2.2 W/m2. The code that calculates the temperature for different values of ECS  is available here. Figure 4 shows the temperature response calculated from the model using AR5 forcing for different  values of ECS.

Comparison of H4 to ECS values ranging from 1.5-4C. The thinbk black line is the 5 year running average of anomaly data

Comparison of H4 to ECS values ranging from 1.5-4C. The thick black line is the 5 year running average of anomaly data

Now looking in more detail at recent temperatures where the cumulative effect of past forcing is strongest, we see how unusual the current hiatus in warming appears.

Comparison of ECS with H4 anomalt details since 1960.

Detailed comparison of ECS with H4 temperature anomalies details since 1960. Thick black line is an  FFT smoothing through the temperature anomaly data with a 5 year filter.

Values of ECS > 3 or ECS < 2 are ruled out by the data. The most likely value for ECS consistent with the recent data is apparently slightly less than  2.5C. The longer the hiatus continues the lower the estimate for ECS.

The overall result from this analysis is ECS = 2.3 ± 0.5 C.  The error is really asymmetric so it is more like  +0.5 and -0.3

Let’s see if all this works out as being consistent with the value of TCR that we measured before, and isolate the energy imbalance \Delta{Q}

\frac{ECS}{TCR} = \frac{\Delta{F}}{\Delta{F}-\Delta{Q}}

\Delta{Q} = \Delta{F}(1 - \frac{TCR}{ECS})

= 0.7 ± 0.5 W/m2

This is consistent with other values for \Delta{Q} .

In summary we have shown that there has been a remarkable approximate agreement between pure CO2 forcing and net CMIP5 forcing. This has allowed us to fit the Hadcrut4 temperature anomaly data to derive a value of TCR = 1.6 ±0.2C. The equilibrium climate sensitivity has been measured by using a model derived value for ocean temperature response to forcing of the form \Delta{T}(t) = \sum_{k=1}^N (\Delta{T}_{0}(1 - e^\frac{-t}{15})) . By integrating each annual pulse of  CMIP5 model forcings, we have compared different values for ECS to the Hadcrut4 anomaly data. This hysteresis effect becomes stronger over time so the current hiatus in warming strongly distinguishes between different values of ECS. Values greater than 3C are ruled out as are values < 2C. The best estimate  for ECS based on this method is 2.3 ± 0.5. The values measured values of TCR and ECS are for a total net forcing of 2.2W/m2 with an energy imbalance of 0.7 ± 0.5.



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