Climate Sensitivity and the 33C Discrepancy

The connection between greenhouse gases such as CO2  and climate change is solid. The global land temperature record is easily on track for a 3° C swing due to a doubling of CO2 levels; this also happens to be the consensus mean for climate sensitivity over a set of peer-reviewed models.

The breakdown of the 3°C doubling is that it is equal parts due to CO2 doubling, water vapor (which comes along for a ride), and a set of miscellaneous positive feedbacks

"If Earth were a blackbody without climate feedbacks the equilibrium response to 4 W/m2 forcing would be about 1.2°C (Hansen et al., 1981, 1984; Lacis et al., 2010), implying that the net effect of all fast feedbacks is to amplify the equilibrium climate response by a factor 2.5. GISS climate models suggest that water vapor and sea ice feedbacks together amplify the sensitivity from 1.2°C to 2-2.5°C. The further amplification to 3°C is the net effect of all other processes, with the most important ones probably being aerosols, clouds, and their interactions. "

 J. E. Hansen and M. Sato, “Paleoclimate implications for human-made climate change,” Climate Change, pp. 21–47, 2012. 

All the pieces to the climate change puzzle are interlocking. Every new piece of evidence has to be consistent with prior understanding, otherwise it weakens the consensus view. I have not been very successfull at finding any real holes in the GHG AGW argument myself. That is likely more do to my lack of detailed knowledge of the intricacies of the models, but it doesn't prevent me from trying other angles. For example, in lieu of my finding any breakthrough debunking arguments, I seek to find simplifications in the overall argument.

This post describes an interesting derivation that ties together the climate sensitivity of CO2 and water vapor along with the 33° C discrepancy between the no-GHG earth and our current average global temperature.

We start with an assumption that the baseline radiatively steady-state temperature is T0 and the temperature raises due to the climate sensitivity:
$$ T = T_0 + \alpha \log(C/C_0) $$
We assume that the second term is mainly due to water vapor, catalyzed by introduction of CO2. This is enough to raise the temperature from a ground-state snowball earth which would occur as the water would normally condense out at 255° K. Thereafter, the water vapor undergoes a bootstrapping process, whereby a fraction of the outgassed vapor serves to feed on itself, and through the GHG process raises the temperature further, leading to more outgassing.

The log dependence of water vapor concentration, C, on temperature, T, reflects the doubling sensitivity of a GHG. Water vapor works much like CO2 in this regard, only that it needs a boost from a non-condensing gas to sustain its elevated concentration.

Next, we assume the outgassing of water vapor follows the Clausius-Clapeyron activation relation, whereby the partial pressure is activated by a Boltzmann factor with an activation energy corresponding to the heat of vaporization.
$$ H = 0.42\:{ eV} = 4873\:{ Kelvins} $$
Then
$$ C/C_0 = \beta e^{-H/T} $$
We can make the assumption that the temperature, T, reaches a steady-state such that the positive feedback limits its extent as the Boltzmann acts as a rather weak feedback term. (This is not runaway warming we are talking about)

When we make the algebraic substitution for C/C0 in the first equation.
$$ T = T_0 + \alpha \log(\beta e^{-H/T}) $$
Because of the log(exp(X))=X identity this reduces to:
$$ T = T_0 + \alpha \log(\beta) - \alpha H / T $$
In terms of solution space, this is a quadratic with respect to temperature. For real valuations, we have a low temperature and high temperature solution.

Graphically, the intersection points look like the following.

Figure 1: Graphical solution to constrained positive feedback.
The red dots show the intersection points.

Mapping this to the real process that is occurring, the low T point likely corresponds to the no-GHG temperature of 255° K, while the high T point is intuitively the steady-state temperature of 289° K. In other words, the gap between the two intersection points is 33° C.
The quadratic equation is
$$ T^2 - (T_0 + \alpha \beta) T - \alpha H $$
with solution
$$ T=\frac{-\gamma\pm\sqrt{\gamma^2-4\alpha H}}{2} $$
where
$$ \gamma = T_0 + \alpha \log(\beta) $$
Safely assuming that the cold and hot solutions are symmetric about the midway point (255+289)/2 :
$$ \Delta T = \pm \sqrt{\gamma^2 - 4 \alpha H}/2 = \pm 16.5° C$$
where
$$ \gamma = T_{low} + T_{high} $$
We can now invert to determine alpha:
$$ \alpha = \frac {\gamma^2 -{2 \Delta T}^2}{4 H} $$
Inserting the values for the 3 fixed values:
$$ \alpha = 15.1° C $$
The climate sensitivity for a doubling of water vapor is
$$ 10.5° C = 15.1 log(2) $$
Finally, we can calculate how much a CO2-induced temperature change will trigger a corresponding rise in water vapor-induced temperature, i.e. it acts as the control knob as Lacis phrased it.

From the differential Clausius-Clapeyron $$ dC/C = {H/T^2} dT $$
with the standard dT for CO2 doubling given by Lacis [1] of 1.23°C. Then dC/C = 0.072 at T=289°K, which gives a sensitivity
$$ \alpha \log(1+dC/C) = 1.05° C $$
for the water vapor rise carried along by the CO2 doubling.
To get to the 3.0° C value we add together 1.23° C from CO2, 1.05° from H20, and approximately 0.7° C from other GHGs (see The NOAA Annual Greenhouse Gas Index) and albedo positive feedbacks.


Comparing to the Berkeley BEST land-based temperature values  with two different sensitivities, one can see how well the consensus value tracks the historical data.

Figure 2: Consensus CO2 GHG model against historical temperature data.
Lower curve is sensitivity of 3°C for a doubling of CO2

---

[1]  A. A. Lacis, G. A. Schmidt, D. Rind, and R. A. Ruedy, “Atmospheric CO2: principal control knob governing Earth’s temperature,” Science, vol. 330, no. 6002, pp. 356–359, 2010.