There seems to have quite a lot of dicussion recently on Watts Up With That (WUWT) about climate sensitivity, in particular the Transient Climate Response (TCR) and the Equilibrium Climate Sensitivity (ECS). Some of what has been claimed on WUWT has got me rather confused, so I thought I would write this post to clarify what I think and also to give others an opportunity to correct me if my understanding is wrong.

I’ll start with the TCR. It is defined here as

the change in the global surface temperature, averaged over a 20-year period, centred at the time of atmospheric carbon dioxide doubling, that is, at year 70 in a 1% per year carbon dioxide increase experiment with a global coupled climate model. It is a measure of the strength and rapidity of the surface temperature response to greenhouse gas forcing.

On a number of WUWT posts, it has been calculated using

TCR = *F*_{2x} Δ*T*/Δ*F*,

where *F*_{2x} is the change in forcing due to a doubling of CO_{2} (typically calculated to be 3.7 W m^{-2}), and Δ*T* and Δ*F* are the measured changes in temperature and forcing over some time interval. Firstly, this kind of makes sense because it tells you, given a change in temperature Δ*T* due to a change in forcing Δ*F*, what the change in temperature will be once Δ*F* = *F*_{2x}. However, it does seem to be an approximation in that it doesn’t depend on the rate of increase of CO_{2} while formally the TCR is defined as the temperature (averaged over 20 years) after the CO_{2} has doubled at a rate of 1% per year for 70 years. Also, it seems like Δ*F* should be the change in forcing due to CO_{2} only. If not, Δ*F* can be larger than that due to CO_{2} alone and the TCR will equal Δ*T* before CO_{2} has doubled (i.e., it will underestimate the TCR).

For example, in this post Willis Eschenbach attempts to calculate the climate sensitivity directly from model results using an equation of the form (I’m leaving out some terms, but this is the basic form)

*T*(*n*+1) = *T*(*n*) + λΔ*F*(*n*+1),

where he refers to λ as the climate sensitivity. However, he seems to use the total change in forcing, Δ*F*, and so this would seem to be the climate sensitivity to an increase in total forcing of 3.7 W m^{-2}, rather than the sensitivity to a doubling of CO_{2}.

The ECS is defined according to this as

a measure of the climate system response to sustained radiative forcing. It is defined as the equilibrium global average surface warming following a doubling of CO2 concentration.

Again, this seems to be determined on a number of WUWT posts using

ECS = *F*_{2x} Δ*T*/(Δ*F* – Δ*Q*),

where the terms are the same as for the TCR and Δ*Q* is the rate at which the oceans are being heated. Again, this makes sense. If, after some time interval, the forcing has changed by Δ*F* with Δ*Q* going into heating the oceans and the difference (Δ*F* – Δ*Q*) going into heating the surface, then the measured change in temperature Δ*T* tells you how much this net forcing increases the surface temperature. Multiplying the by change in forcing caused by a doubling of CO_{2} then tells you how much the temperature will eventually increase by if CO_{2} will double.However, this again seems like an approximation and that to determine the ECS properly probably needs detailed climate modelling.

Also, it seems like Δ*F* and Δ*Q* should, again, only be the contributions due to CO_{2}. If not, then it would seem like the ECS is an estimate of the change in temperature if the total forcing changes by 3.7 W m^{-2}, rather than if the CO_{2} concentration doubles. I would be interested to know if my understanding is correct, so if anyone knows more about this than me (of which there must be many) I’d be keen for them to clarify. It also seems, if my understanding is roughly correct, that the reason a number of the posts on WUWT are getting lower climate sensitivities than other work is because they’re using the wrong values for Δ*F* and Δ*Q* – although it would be good if someone could clarify whether this is the case or not.

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Okay, just for clarity I have established that I think my understanding was incorrect. I had assumed that

F_{2x}was that change in forcing due only to a doubling of the CO_{2}concentration. This, I think, is not correct. It is the change in all known forcings after CO_{2}concentrations have doubled and hence the assumptions made by Nic Lewis and Otto et al. (2013) are correct. This has still been useful (for me at least) as I now think I understand this reasonably well – although I’ve thought that before and been wrong then 🙂Pingback: Another Week of Anthropocene Antics, May 26, 2013 – A Few Things Ill Considered