Calculation of No-Feedback Temperature Increase due to CO2 Doubling

There are many places, e.g., https://en.wikipedia.org/wiki/Climate_sensitivity, where you can read statements such as, "Without any feedbacks, a doubling of CO₂ (which amounts to a forcing of 3.7 W/m²) would result in 1 °C global warming, which is easy to calculate and is undisputed. The remaining uncertainty is due entirely to feedbacks in the system, namely, the water vapor feedback, the ice-albedo feedback, the cloud feedback, and the lapse rate feedback" (Rahmstorf, Stefan (2008). "Anthropogenic Climate Change: Revisiting the Facts". In Zedillo, E. Global Warming: Looking Beyond Kyoto (PDF). Brookings Institution Press. pp. 34–53.) However, finding the calculations by which this 1.0 - 1.2 °C is determined is not easy (https://judithcurry.com/2010/12/11/co2-no-feedback-sensitivity/).

In fact, the no-feedback temperature increase from doubling CO₂ is easy to calculate. Two equations are needed to determine the increase: the Arrhenius relationship which determines the increase in radiative forcing (∆RF) due to a CO₂ increase:

∆RF = α•ln((C₀+∆C)/(C₀)) (Eq. 1)

where ∆RF is the change in radiative forcing, C₀ is the initial concentration of CO₂, ∆C is the concentration change, and α is generally accepted as 5.35, and the following arrangement of the Stephan-Boltzmann Law:

((RF₀+∆RF)/RF₀)^1/4 = ((T₀+∆T)/T₀) (Eq. 2)

where RF₀ is the initial radiative forcing for temperature T₀, ∆RF is defined above, and ∆T is the temperature change.

For a doubling of CO₂ Arrhenius does indeed gives 5.35•ln(2) = 3.7 W/m² as the ∆RF mentioned above. To use Equation 2, note that T₀ is the current temperature of Earth, 288K. To determine the appropriate RF₀ for this temperature we can use the no-greenhouse model of the Earth which is used to determine a 33° C warming due to these gases. In this model, the ~1360 W/m² solar radiant flux when the sun is directly overhead is used to calculate an average flux over the entire planet, after accounting for an albedo of 0.3, of 240 W/m². The full Stephan-Boltzmann Law (https://en.wikipedia.org/wiki/Stefan-Boltzmann_law) is then used to calculate a surface temperature of 255K. Equation 2 can then be used to determine that an additional 150 W/m², or 390 W/m² altogether, is needed to achieve T₀.

We can now fill in all the needed values in Equation 2, and solve for ∆T:

((390+3.7)/390)^1/4 = ((288+∆T)/288), ∆T = 0.68K

0.68K?  What happened to the 1K rise that has been publicized so widely?  Is there an error in my calculations, or the premises which set them up?

As noted above, "finding the calculations by which this 1.0 - 1.2 °C is determined is not easy."  A possibility does occur to me, however.  If instead of using the current planetary values for T₀ and RF₀ we use the values calculated in the no-greenhouse model, the results of solving Equation 2 are somewhat different:

((240+3.7)/240)^1/4 = ((255+∆T)/255), ∆T = 0.98K

0.98K is remarkably close to the "undisputed" value for the no-feedback increase.  To continue in this vein, if the actual current global temperature is used instead of the no-greenhouse value, the rise "rises" further, to 1.1K.

There are other possibilities.  For example, the value of 5.35 used in Equation 1 is, I assume, a global average derived from various locations where the water vapor content of the air at those locations are significantly different -- which they can be by as much as 300 fold.  This is important because, as water vapor absorbs most of the thermal IR given off from the ground, CO₂'s absorbance will be either greater or smaller as a consequence (this is why AGW should be stronger at the poles than the equator).  Use of a single, global temperature, rather than summing results from regions colder or hotter, may have an effect as well.

These complications can be explored in another post.  For the moment, I will accept the 0.68K value calculated here as accurate, and ask the next logical question:  how much of this no-feedback warming should we expect so far?  By so far, I mean going back about 150 years when CO₂ levels were 280 ppm and global temperature approximately 1K cooler.  Since current (2017) CO₂ levels are given as 405 ppm -- a 44.6% increase -- we can use Equation 1 to calculate a ∆RF of almost 2.0 W/m².  Equation 2 converts this into a ∆T of 0.37K, or only 37% of the full estimated temperature rise during the last century and a half.

I can't resist pushing this article beyond the scope of its title.  What, for example, might be the total no-feedback temperature effect of all radiative forcings over the last century and a half taken together.  To answer this I include the following chart:

There is considerable uncertainty in many of these measurements, but if the average values are reasonably correct, they appear to cancel each other out, leaving the CO₂ forcing close to the average.

The remaining factor to be considered is the climate sensitivity factor.  This is difficult to determine with confidence, and the literature shows a wide range of values, from below 1 to as much as 4 or 5.  The IPCC offers a range of 1.5-4.0, but values lower than this are regularly reported.  If we accept a value of around 1.4 however (https://judithcurry.com/2016/04/25/updated-climate-sensitivity-estimates/), an interesting suggestion emerges from the data.  If the CSF is ~ 1.4, then the total anthropogenic warming since the 1860s is about a half of a kelvin, or about one half the total estimated warming since that time.  This of course implies a natural warming over this period of about the same.  Again, citing the IPCC, this fraction is at the lowest end of its estimate of 95% certainty that anthropogenic warming is greater than 50%.

It may be even lower than that, however.  According to measurements of radiative forcing due to CO₂ increase over the period of 2000-2010 (https://www.nature.chttps://www.nature.com/articles/nature14240om/articles/nature14240) the increase of CO₂ concentration from 370 ppm to 392 ppm during this period was accompanied by a radiative forcing increase of 0.2 W m⁻² per decade.  Using this data Equation 1 can be rearranged to solve for α:

 α = ∆RF / ln((C₀+∆C)/(C₀)) (Eq. 3)

to yield 3.46, or about 2/3'rds the standard accepted value.  This in turn leads to a no feedback
∆T of 0.25K instead of 0.37K, implying (if we maintain the climate sensitivity of 1.4) a total anthropogenic warming of only 0.35K, or 35% of total warming over the last 150 years.