How Feedbacks Amplify Global Warming

In a previous post (see that post for the cited literature), I calculated ECS from data provided in the Indicators of Global Climate Change for 2025. I made 4 calculations using two different baselines (1850-1900 and 1790-1839) and two different means for current temperature (the last 4 years and the last 12 years). I'd like to share again just one of these so I can illustrate how feedbacks amplify a warming signal. These are the values I used that was based on a ΔT = 1.23°C (the difference between 2013-2025 and 1850-1900).

ΔT = 1.23°C (HadCRUT5)

ΔF = 2.60 W/m² (IGCC25)

EEI = 1.12 W/m² (IGCC25)

ΔF2xCO₂ = 3.93 W/m² (AR6)

For the sake of brevity, I'm going to relabel the forcing for doubling CO2 that I had labeled ΔF2xCO₂ as ΔFc. Using these values and the Energy Balance Equation, we can calculate ECS to be:

λ = ΔT/(ΔF - EEI)

λ = 1.23/(2.60-1.12) = 0.83°C/W/m²

ECS = λ*ΔFc = 0.83*3.93 = 3.3°C

To calculate ECS, we didn't have to know any of the feedbacks. But we do know that there must be positive feedbacks because Planck sensitivity (Sp) is only about 1.2°C, and we've already seen that much warming with just over a 50% increase in CO₂. But the Planck Feedback (λp) is important here, and we can calculate it (when T = 288 K) as:

λp =  -0.4*4σT³ = 3.25 W/m²/°C. 

With no other operating feedbacks, 2xCO₂ would lead to an increase of ΔFc/λp =3.93/3.25 = 1.21°C warming, and we've already exceeded that. However, we can estimate the total size of those feedbacks without knowing the value of any of the individual feedbacks. If ECS = 3.3°C, we can first multiply ECS by Planck feedback:

λp*ECS = 3.25*3.3 = -10.73 W/m²

If we subtract ΔFc from λp*ECS then total forcings from other feedbacks are 10.73 - 3.93 = 6.8 W/m², and that means the total feedback (outside of the Planck Response) is:

λt = 6.8/3.3 = 2.06 W/m²/°C

But how does this value compare to estimates of individual feedbacks? Here is an accounting of the dominant feedbacks in the climate system. Notice that these total a value very similar to what we calculated above:

Water Vapor (λwv): 1.80 W/m²/°C

Lapse Rate (λlr): -0.50 W/m²/°C

Surface Albedo (λsa): 0.35 W/m²/°C

Clouds (λcl): 0.43 W/m²/°C

Total (λt): 2.08 W/m²/°C

Calculating ECS from these value is pretty easy. First, we calculate the total feedback with the Planck Response and then divide ΔFc by the sum:

 λpt = λp + λt = -3.25 + 2.08 = -1.17 W/m²/°C

ECS = -ΔFc/λpt = 3.93/1.17 = 3.36°C

This value matches closely to the calculation above, and it's confirming evidence that indicates that an ECS near 3°C. But we can also think of this in terms of gain vs feedback. Let's define gain (G) as 1/(1-f) where f is the feedback factor. We can calculate f as λt/λp. Notice this is a unitless value, but as f approaches 1, G explodes. But the calculation gives us the same result:

G = 1/(1-f) = 1/(1-λt/λp)

G = 1/(1-2.08/3.25) = 2.78

This means that total feedbacks amplify Planck sensitivity (Sp) by 2.78x. We calculated Sp earlier to be 1.21°C, so we can calculate ECS to be:

ECS = Sp*G = 1.21*2.78 = 3.36°C

The fact that G explodes as f approaches 1 explains why the uncertainty ranges for ECS are larger above the central estimate than below it. You can see this in the plot of G(f) below:

The IPCC's central estimate for ECS is 3°C with a likely range of 2.5°C to 4°C. The range is 2x larger above the central estimate than below it. This is because G is a curve that is somewhat flat when f is very small or negative, but it explodes as you get closer to 1, meaning that adding positive feedbacks multiplies Sp by larger and larger values, and uncertainties with positive feedbacks make ECS values higher than the central estimate more likely. That is, with a central estimate of 3°C, ECS is more likely to be 3.5°C than it is to be 2.5°C.

Of the above feedbacks the cloud feedback (λcl) has been the most difficult to quantify. While the best evidence we have is that it's positive, the uncertainty range is significant at 0.43 ± 0.35 W/m²/°C. However, even if this estimate is wrong and λcl turns out to be negative, that only indicates that some other feedbacks must be more strongly positive than indicated, since warming has occurred to rapidly for the the total λt to be much less than 2 W/m²/°C. This is one reason why scientists can rule out low values of ECS below about 2.5°C. Warming rates are simply too strong for ECS to be low.