From a previous post (Sensitivity of Global Temperature), I was able to detect the differential CO2 sensitivity to global temperature variations. The correlation of temperature anomaly against d[CO2] is very strong with zero lag and a ratio of about 1 PPM change in CO2 per degree temperature change detected per month.

Now, this does not seem like much of a problem, as naively a 1 degree change over a long time span should only add one PPM during the interval. However, two special considerations are involved here. First, the measure being detected is a differential

*rate*of CO2 production and we all know that sustained rates can accumulate into a significant quantities of a substance over time. Secondly, the atmospheric CO2 has a significant adjustment time and the excess isn't immediately reincorporated into sequestering sites. To check this, consider that a slow linear rate of 0.01 degree change per year when accumulated over 100 years will lead to a 50 PPM accumulation,

*if the excess CO2 is not removed from the system*. This is a simple integration where f(T(t)) is the integration function :

$$ [CO2] = f_{co_2}(T(t)) = \int^{100}_0 0.01 t\, dt = \frac{1}{2} 0.01 * 100^2 = 50 $$

The sanity check on this is if you consider that a temperature anomaly of 1 degree change held over 100 years would release 100 PPM into the atmosphere. This is simply a result of Henry's Law applied to the ocean. The ocean has a large heat capacity and so will continue outgassing CO2 at a constant partial-pressure rate as long as the temperature has not reached the new thermal equilibrium. (The CO2 doesn't want to stay in an opened Coke can, and it really doesn't want to stay there when it gets warmed up)

So, if we try the impulse response we derived earlier (Derivation of MaxEnt Diffusion) to this problem, with a characteristic time that matches the IPCC model for Bern CC/TAR, standard:

As another sanity check, the convolution of this with a slow 1 degree change over the course of 100 years will lead to at least a 23 PPM CO2 increase.

Again, this occurs because we are far from any kind of equilibrium, with the ocean releasing the CO2 and the atmosphere retaining what has been released. The slow diffusion into the deep sequestering stores is just too gradual while the biotic carbon cycle is doing just that, cycling the carbon back and forth.

So now we are ready to redo the model of CO2 response to fossil-fuel emissions (Fat-Tail Impulse Response of CO2) with the extra positive feedback term due to temperature changes. This is not too hard as we just need to get temperature data that goes back far enough (the HADCRUT3 series goes back to 1850). So when we do the full combined convolution, we add in the integrated CO2 rate term f(T), which adds in the correction as the earth warms.

$$ [CO2] = FF(t) \otimes R(t) + f_{co_2}(T(t)) \otimes R(t) $$

When we compute the full convolution, the result looks like the following curve (baseline 290 PPM):

The extra CO2 addition is almost 20 PPM just as what we had predicted from the sanity check. The other interesting data feature is that it nearly recreates the cusp around the year 1940. The previous response curve did not pick that up because it is entirely caused by the positive-feedback warming during that time period. The effect is not strong but discernible.

We will continue to watch how this plays out. What is worth looking into is the catastrophic increase of CO2 that will occur as long as the temperature stays elevated and the oceans haven't equilibrated yet.

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