This post was partly motivated by Michael Tobis’s recent post in which he rebuts a doom-monger by pointing out that feedbacks can’t really produce some kind of runaway process in the next few decades, and by my uncertainty as to whether or not energy budget estimate for the ECS properly account for feedbacks. In a simple sense they do (I was just confused) but that doesn’t mean that there aren’t other issues with these estimates.
The standard energy budget method for estimating the transient climate response (TCR) and equilibrium climate sensitivity (ECS) is
where F2x is the change in anthropogenic forcing after a doubling of CO2, ΔT is the change in temperature over the time interval considered, ΔF is the change in anthropogenic forcing over that same time interval, and ΔQ is the system heat uptake rate (i.e., the current energy imbalance). The TCR equation has always made sense to me (if we know how much the temperature has changed for a given change in anthropogenic forcing, then we can estimate how much it should change when the change in forcing is equivalent to that from a doubling of CO2). The ECS equation essentially uses the current heat uptake rate to estimate the warming in the pipeline for the change in anthropogenic forcing, but I’ve always been unsure as to whether or not it properly accounts for the feedbacks due to the warming in the pipeline.
Being a simple physicist, I thought I would redo the energy budget estimate more crudely. All the values I’ll be using are from the top row of Table S2 in Otto et al. (2013)’s Supplementary Information. If the temperature changes by an amount ΔT, then the outgoing flux should increase by
where the 0.62 is an estimate of the surface emissivity. Using Teff = 288 K, and ΔT = 0.75 K gives ΔFout = 2.53 Wm-2. If the heat uptake rate (energy imbalance) is 0.65 Wm-2, then that means that the net change in radiative forcing is 3.2 Wm-2. If the change in anthropogenic forcing over the same time interval is ΔF = 1.95 Wm-2, then that means feedbacks must be providing a radiative forcing of ΔFfeed = 1.23 Wm-2.
If we assume that the feedback forcing depends linearly on ΔT (which seems reasonable in this simple approximation) then we can assume it depends linearly on ΔFout (which is okay for small changes in ΔT). In the absence of feedbacks, a doubling of CO2 would – for equilibrium – require an increase in outgoing flux of ΔFout = 3.44 Wm-2. Therefore, if ΔFfeed = 1.23 Wm-2 when ΔFout = 2.53 Wm-2, the feedback will be ΔFfeed = 1.67Wm-2 when ΔFout = 3.44Wm-2. However, as the temperature rises in response to this feedback, it will produce an additional feedback. As it rises in response to this additional feedback, it will produce another additional amount of feedback. Essentially it is an infinite sum. If F is the ratio ΔFfeed / ΔFout, the net effect of feedbacks will be
If F < 1, then this converges. In the example I've given here, F = 0.49. You can calculate the sum yourself, and should get 1.96. Therefore, at equilibrium, the influence of feedbacks means that the total change in forcing (anthropogenic + feedbacks) should be 1.96 x 3.44Wm-2 = 6.74Wm-2. If I now use the second equation in this post with Fout = 6.74Wm-2, and solve for ΔT, I get 2 K. So, exactly the same as Otto et al. (2013). So, it seems my concern that the energy budget estimate for ECS didn’t properly account for feedbacks was wrong, but I think I’ve shown that you can get the same result by explicitly considering feedbacks (which rather scuppers those who keep claiming that feedbacks have yet to manifest themselves). It does, however, nicely illustrate how the energy budget estimate for the ECS very explicitly assumes that feedbacks are linear.
Having done this, I realised that I can now constrain one more parameter in my two-box model, and that is the relationship between temperature change and feedback forcing. In that model I used Ffeed = 1.8ΔT, whereas this analysis suggests it should be Ffeed = 1.6ΔT. If I use the latter in my two-box model, the only free parameter now is β which I adjust slightly (β = 0.2) so as to recover a good fit. With these parameters, I get a TCR of 1.6 K and an ECS of 2 K (as one might expect).
Anyway, there’s not really much to this post, other than my attempt to check whether or not energy budget estimates properly account for feedbacks (it seems that they do). I should add that I’ve completely ignored any error analysis or uncertainties, so I’m not claiming that these estimates are reasonable values, simply that if I use a consistent set of values I can recover the same ECS if I use the standard energy budget method, my feedback method (which is the same as the energy budget method, but just written out in a way I can understand) and my two-box model. When I did play around with some of the values, the results can change quite substantially which may indicate how sensitive such methods are to the values used, but I haven’t done some kind of detailed study to really address this. Anyway, as usual, feel free to correct my working/thinking, through the comments.