Is Happer Right that Warming by CO2 is Too Small to Matter?

In a recent talk (relevant excerpt here) given to an Australian political group called the Institute for Public Affairs (IPA), William Happer argued that doubling CO2 causes only 0.71 K warming, and that amount of warming for 2xCO2 is too small to matter. He then suggests that in order to make CO2 a problem, scientists had to invent giant feedbacks to amplify warming by as much as 10x the amount caused by CO2 alone. I've seen this claim repeated by others on X and other social media platforms, but as best I can tell Happer originates this particular argument. So I'd like to consider, is this plausible at all? I think it's pretty easy to investigate this and show conclusively that it is not. In fact, even Happer disagreed with this claim as recently as 2020.

Happer's Argument

At about the 1 minute mark of the above linked excerpt, Happer explains his math on how he arrives at 0.71 K for equilibrium climate sensitivity (ECS). He begins with the equation:

F = εσT^4, where

ε is emissivity

σ is the Stefan-Boltzmann Constant 

T is GMST = 288.7 K

He then asserts that 2xCO2 changes emissivity ε by "about -1%" (Δε/ε = -1/100). To keep F constant for an equilibrium change in temperature, therefore, simple calculus using the above equation and its derivative lets us write:

0 = ΔεσT^4 + εσ4ΔT^3

Rearranging terms and simplifying, we can derive the equation:

ΔT/T = -Δε/4ε

And solving for ΔT, we get

ΔT = -T*Δε/4ε = 288.7(1/4)*(1/100) = 0.72 K

Let's cut him some slack on arriving at 0.71 K instead of 0.72 K. Perhaps there were figures he used with more significant digits than reported that led to his 0.71 K figure. His math is essentially reproduceable if we accept his emissivity values. From here he argues that Le Chatelier's Principle means that there are no significant positive feedbacks amplifying warming, so he concludes 1) that ECS is essentially the same as the pre-feedback sensitivity of 0.71 K and 2) that climate scientists have no reason to add positive feedbacks to this value other than to scare people.  Let's look at these claims in turn.

Pre-Feedback Sensitivity

Happer's calculations above depend on the value for Δε/ε being "about -1%." But how close to -1% is "about -1%?" We can actually figure this out if we look at a similar calculation he used in a 2020 manuscript he put on the arXiv[1] and a political document he wrote for the CO2 Coalition. Here Happer again used the equation F = εσT^4 and its derivative, δF/δT = 4εσT^3, and according to him, we can rearrange and combine terms to allow for this equation:

ΔT = T*ΔF/4F

This means necessarily that -Δε/ε = ΔF/F, and if we use Happer's value from his 2020 manuscript[1] for ΔF = 3 W/m^2 and the value from his CO2 Coalition document of F = 277 W/m^2, then ΔF/F  = 1.1%. This is qualifies as being "about 1%." However, these values are almost certainly wrong. His ΔF is almost certainly too low; calculations almost always put ΔF between 3.7 and 4 W/m^2, and the IPCC AR6 currently uses 3.93 W/m^2. His value for F is similarly too high; F at equilibrium must equal the incoming solar flux, which is:

F = S*(1-α)/4, where

S is TSI = 1361 W/m^2

α is albedo = 0.3

F = S*(1-α)/4 = 1361*(1-0.3)/4 = 238 W/m^2

If we use these more accurate values for ΔF and F, then ΔF/F = 1.6%, which is a little less close to "about 1%" but still in the ballpark. But let's see what happens to Happer's calculation if we use correct values for ΔF/F:

ΔT = T*ΔF/4F = 288.7*3.93/238 = 1.2 K

Just using correct values for ΔF/F increases Happer's original result by about 70%. And we can actually confirm the IPCC's value for pre-feedback sensitivity by doing a calculation which incorporates the Planck Response (λp). We can simply take the ΔF for 2xCO2 and divide it by λp = 3.27 W/m^2/K at 288.7 K.

ΔT = ΔF/λp = 3.93/3.27 = 1.2 K

And 1.2 K is consistent with standard pre-feedback sensitivity found in the scientific literature. It would seem that Happer is using a deliberately imprecise value for -Δε/ε to give the false impression that pre-feedback sensitivity is lower than the standard pre-feedback sensitivity found in the scientific literature. Almost certainly 0.71 K is too low, and using correct values for ΔF/F and -Δε/ε values we get a value that is closer to 1.2 K for 2xCO2.

Le Chatelier's Principle
But what about Happer's claim that there are no positive feedbacks to amplify this pre-feedback sensitivity? Happer claims that "it's almost impossible to justify this idea of a positive feedback, but you need it, otherwise CO2 is too wimpy to be worried about." Happer's reasoning for this is the Le Chatelier's Principle, which he summarizes by saying that "feedbacks are negative... positive feedbacks almost never occur." He then says that scientists just assume positive feedbacks because if they don't they won't get money for their laboratories. This "argument" is frankly bizarre, and I don't see how even Happer believes it. I strongly suspect he absolutely knows both that he's wrong and why he's wrong. He he appears to be saying this in attempt to pull the wool over the eyes of those who aren't going to check up on his claims.

Le Chatelier's Principle is simply this: "If the equilibrium of a system is disturbed by a change in one or more of the determining factors (as temperature, pressure, or concentration) the system tends to adjust itself to a new equilibrium by counteracting as far as possible the effect of the change." More simply stated, if you perturb a system at equilibrium, the system tends to achieve a new equilibrium state. And this is precisely what we observe if there are positive feedbacks that push ECS to be in to the neighborhood of 3 K. When the climate system is perturbed by a 2xCO2 forcing of 3.93 W/m^2, positive feedbacks can amplify that warming, but the Planck Response is actually a very strong negative feedback that returns the climate system to a new equilibrium state at a higher temperature. Whether ECS is 0.72 K or 3 K or 10 K, a perturbation of the climate system doesn't cause the system to spiral into runaway warming. The Planck Response ensures that a new equilibrium temperature is achieved. The central estimate for this is that 2xCO2 causes that new equilibrium temperature to be reached at a ~3 K warmer temperature. There is absolutely no violation of Le Chatelier's Principle here. It is clearly and obviously observed, and Happer knows this. He knows about the Planck Response, and he knows that the E in ECS stands for "equilibrium." 

We have enough empirical data about forcings and temperature to do a simple calculation for the value of ECS. Using non-controversial values for forcings and temperature with the energy balance equation, we can do a rough, back of the envelope calculation. Here's the energy balance equation solved for sensitivity (λ):

λ = ΔT/(ΔF - EEI), where

ΔT is the change in temperature above the 1850-1900 mean = 1.25 K.
ΔF is total forcings = 2.7 W/m^2
EEI is the earths' energy imbalance = 1.1 W/m^2

λ = 1.25/(2.7-1.1) = 0.78 K/W/m^2

If we multiply this value by the forcings for 2xCO2, we get

ECS = λ*ΔF2xco2 = 0.78*3.93 = 3.07 K

Of course there are uncertainties in all of these values that could push this result up or down, but none of these values for ECS  would violate Le Chatelier's Principle, as Happer claims. If this calculation is correct, then positive feedbacks amplify pre-feedback sensitivity by 2.5x, but the Planck Response still makes it so that a new equilibrium temperature is reached. Scientists generally believe these positive feedbacks are operating in the climate system because 1) they are observed (the water vapor feedback is well documented in the literature) and because 2) warming is occurring too rapidly, given ΔF and EEI estimates we have, for ECS to be lower than about 2.5 C.

Conclusion

In another post, I describe the physics involved with feedbacks in a little more detail.  But in 2020, even Happer admits that positive feedbacks are real and ECS could be in the neighborhood of 2.3 K. In fact, we can quote him saying this: "Doubling CO2 concentrations with water vapor feedback increases the surface temperature warming to ∆θo = 2.3 K from ∆θo = 1.4 K, or by a factor of 1.6." So here Happer shows both that pre-feedback sensitivity can be higher than 1 K and that water vapor acts as a positive feedback that amplifies warming by 1.6x; there's no violation of Le Chatelier's Principle. In other words, in 2020, Happer was able to clearly articulate that he's currently wrong and exactly why he's currently wrong in this lecture he gave to the IPA. 

References:

[1] W. A. van Wijngaarden and W. Happer. Dependence of Earth’s Thermal Radiation on Five Most Abundant Greenhouse Gases. Unpublished manuscript. 2020.
https://arxiv.org/pdf/2006.03098.pdf