The Physics of the Climate Response to Doubling CO2

In a recent post, I provided a standard calculation for the effective temperature of Earth (the earth without GHGs) to be 255K. This calculation used the temperature and radius of the sun, the earth's distance from the sun, an emissivity of 1 and albedo of 0.3. So, if the earth had no atmosphere, the average surface temperature of the earth would be about -18 C. However, we know that currently, the earth is about 33 C warmer than this; it's ~288 K or 15 C. The 33 C difference is caused by the greenhouse effect, and the above diagram of the Earth's emission spectrum illustrates how this is accomplished. Greenhouse gases slow down upwelling LW radiation escaping into space, trapping heat in the troposphere, such that the surface warms in response.

                                Climate at Equilibrium to 288 K

Current Conditions

I decided it would be fun to explain how this works given the physics accounting for various fluxes that influence the Earth's surface temperature. What I want to do is describe current climate physics at a GMST of 288K and then consider the impact of doubling CO2 with an ECS of 3 C. For the following, I will use longwave (LW) surface emissivity (ε) of 0.98. If we take current temperature to be 288 K, then we can calculate the upwelling surface flux of LW radiation (Fs) as

Fs = εσT^4 = εσ*288^4 = 382.3 W/m^2 (1)

The outgoing top of the atmosphere flux (Fout) and incoming solar radiation (Fin) are nearly identical - the difference between the two is the earth's energy imbalance, currently about 0.8 W/m^2. The value of incoming solar flux can be calculated (where α is albedo and S = TSI) as

Fin = (1-α)*S/4 = (1-0.306)*1361/4 = 236.1 W/m^2 (2)

At equilibrium, Fin and Fout are equal to each other at 236 W/m^2, so the difference between the values from (1) and (2) is the greenhouse effect (G). This effect is the amount of heat radiated from the surface doesn't escape into space. That is, if the atmosphere were free of greenhouse gases, Fs would equal Fin and Fout. So the greenhouse effect can be calculated as 

G = Fs - Fout = 382.3 - 236.1 = 146.2 W/m^2 (3)

The ratio G:εσT^4 is the normalized greenhouse effect (g). While technically this value increases with temperature, it is treated as a constant, so as GHGs increase in the atmosphere (within reason), g remains roughly the same. So

g = G/εσT^4 = 146.2/382.3 = 0.4 (4)

Consistent with the IPCC[2]. I'm holding g at a constant 0.4. If we differentiate the Stephan-Boltzmann equation, we can use this to calculate the Planck Response, which allows us to account for the fact that warmer objects radiate more than colder objects. That is to say, as the planet warms, there's an increase increase in planetary infrared energy loss that is vertically uniform on the surface and in the troposphere; it is this increase in surface radiation that increases Fout until it matches Fin again. The Planck Response is is a negative feedback and can be calculated as:

λp = (g-1)*4σT^3 = -0.6*σ288^3 = -3.25 W/m^2/C (5)

The values from these 5 equations are non-controversial, though their precise values can be better estimated by other means than the simple equations I'm using above. Some other accountings for these values do not include surface emissivity (ε), so the Fin, Fout, and Fs values would be slightly higher than what I've calculated above. Including emissivity makes these values a little more accurate, but the results aren't that much different those that don't include emissivity.  This is just a thought experiment, though, and what I want to do here is simply to show how the math works so that when there's a perturbation of the Earth's energy imbalance from a doubling of CO2, the earth warms until it reaches a new equilibrium at a higher temperature. I want to show how this works using standard estimates for ECS = 3 C and positive feedbacks (λt) from water vapor, clouds, etc. without leading us into a runaway greenhouse effect like what we see on Venus.  

                          Climate at Equilibrium to 291 K (2xCO2 with ECS = 3 C) 

Doubling CO2 Before Feedbacks

Above I showed the incoming flux of solar radiation (Fin) to be 236 W/m^2, and if the Earth was at equilibrium with no greenhouse effect, the flux escaping the earth would be identical to that value. Using the Earth's effective temperature (255 K), and the Stefan-Boltzmann equation, we can calculate

Fout = εσT^4 = εσ*255^4 = 235 W/m^2

This is roughly equivalent to the Fin calculation above, confirming that without the influence of greenhouse gases, the Earth's temperature would be stable at about 255 K.

A perturbation of the Earth's Energy Imbalance (EEI), in this case, from doubling CO2 (ΔFco2), causes 5.35*ln(2) = 3.7 W/m^2 increase in radiative forcing - more energy enters the climate system than escapes into space. This means that while Fin remains the same, Fout decreases by 3.7 W/m^2. The surface of the planet must warm until the imbalance is restored to 0. 

There are a couple ways we can estimate the sensitivity of this change in EEI before assessing the impact of feedbacks. First, we can divide the forcing for doubling CO2 by the Planck Response. This yields 

ΔFco2/λt = 3.7/3.25 = 1.14 C 

Second, we can combine the Stefan-Boltzmann equation with its derivative[4]. If we do that with the Earth's current temperature (288 K) we get a very similar result:

ΔT/T = (1/4)*ΔFco2/F, so

ΔT = T(ΔFco2/4F) = 288*(3.7/4*236.1) = 1.13 C

These two calculations are roughly equivalent to each other, and they are also consistent with what is produced in model calculations.

Doubling CO2 After Feedbacks

As the surface of a planet warms, it radiates more toward space than when it was cooler. This is again expressed in what is called the Planck Response or Planck Feedback (λp), which, as you can see from equation (5), increases with temperature. As greenhouse gases increase in the atmosphere, these GHG forcings are amplified by positive feedbacks and dampened by negative feedbacks. The total of these feedbacks here is expressed here with λt. However, λt is always smaller than -λp, and the Planck Feedback increases with temperature, so perturbations of EEI can always be restored to a new equilibrium temperature.

The energy imbalance caused by doubling CO2 requires that a new, higher equilibrium temperature be achieved by increasing the upwelling surface LW flux (Fs) - that is, the surface temperature must increase. With ECS = 3 C, doubling CO2 increases surface temperature from 288 K to 291 K. The resulting Fs would thus become

Fs = εσT^4 = εσ*291^4 = 398.5 W/m^2

This increases the greenhouse effect (G) to  398.5 - 236.1 = 162.4 W/m^2, assuming constant albedo. That's an increase of 162.4 - 146.2 = 16.2 W/m^2 to the greenhouse effect. Because the Earth's surface is now at a higher temperature, the Planck Response must also increase. The Planck Feedback (λp) at 291 K is

λp = (g-1)*4σT^3 = -0.6*σ291^3 = -3.35 W/m^2/C. 

We can multiply this value by ECS = 3 C to get a total of -3.35*3 = -10 W/m^2 for the Planck response for this perturbation of the Earth's energy imbalance. This means that the total of all feedbacks λt need to account for the difference between ΔFco2 and -λp*ECS. Total feedbacks (λt) are estimated to produce an additional 2.1 W/m^2/C[3] so λt*ECS = 6.3 W/m^2. And if you add up all three of these values, you get 0 W/m^2.

λp*ECS + ΔFco2 + λt*ECS = 0

-10 W/m^2 + 3.7 W/m^2 + 6.3 W/m^2 = 0 W/m^2 

Conclusion

This means that a new equilibrium will have been reached at 291 K when EEI is reduced to 0. This new equilibrium is reached at 291 K if ECS = 3 C, λp = -3.35 W/m^2/C at the resulting 291 K, and λt = 2.1 W/m^2/C. However, there are other combinations of values that would also add to 0 at a different equilibrium temperature. Increasing ECS would also increase -λp and λt. Likewise, decreasing ECS would decrease -λp and λt.  For various ECS values in the IPCCC's range, here are the results you would get:

        For ECS = 2.5 C        λp = -3.34 W/m^2/C        λt = 1.85 W/m^2/C

        For ECS = 3.0 C        λp = -3.35 W/m^2/C        λt = 2.12 W/m^2/C

        For ECS = 3.5 C        λp = -3.37 W/m^2/C        λt = 2.31 W/m^2/C

        For ECS = 4.0 C        λp = -3.39 W/m^2/C        λt = 2.46 W/m^2/C

So this isn't proof that ECS = 3 C. But it does show that runaway warming like Venus is not really something to be concerned about, since this would require that at some point λt > λp and from that point λt increases with temperature at least as rapidly as λp increases. But this is not at all likely. Perhaps more significantly, it also shows that to the extent we can constrain estimates for λt, we can also limit reasonable values for ECS. Likewise, constraining our estimate of ECS would also limit reasonable values for λt. The values for ECS, λt and λp cannot just be anything. They are related to each other and bound by known laws of physics. Constraining these values will continue to improve model simulations and predictions, but the the constraints that the laws of physics place on these values make it possible to understand the climate-related response to these perturbations to make reliable assessments of how the Earth's climate system will respond to doubling CO2.

References:

[1] Gavin Schmidt, "The CO2 problem in 6 easy steps." RealClimate. Aug 6, 2007. https://www.realclimate.org/index.php/archives/2007/08/the-co2-problem-in-6-easy-steps/

[2] IPCC AR6 WG1. https://www.ipcc.ch/report/ar6/wg1/

[3] Sherwood, S. C., Webb, M. J., Annan, J. D., Armour, K. C., Forster, P. M., Hargreaves, J. C., et al. (2020). An assessment of Earth's climate sensitivity using multiple lines of evidence. Reviews of Geophysics, 58, e2019RG000678. https://doi.org/10.1029/2019RG000678

[4] This formula can be derived from F = εσT^4 and its derivative, δF/δT = 4εσT^3. Reorganizing terms we get εσ=F/T^4 and εσ=δF/δT*4T^3, and combining equations gives us δF/4F = δT/T.