Where Has All the Carbon Gone?

This post covers a very important topic which is a new interest of mine.  Understanding how climate works is a series of layers, in this case DeWitt discovered some interesting discrepancies between atmospheric carbon models and measured results.

Guest post by DeWitt Payne.

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Where Has All the Carbon Gone?

(Sung to the tune of Where Have All the Flowers Gone?)

Introduction

I want to thank Harry (comments #22 #39 in the Mike Hulme – Consensus Science thread) for getting me started on this. This was going to be a short post, but the more I thought about it, the more I needed to explain.

Carbon is going missing.  We know to a reasonable approximation how much carbon is being emitted to the atmosphere by burning fossil fuel and land use changes.  But the concentration in the atmosphere isn’t going up as fast as expected.  Where is it going and what will be the long term effect?  I don’t have the answer, but I’ve learned some things by looking at the data that weren’t obvious in relation to what I’ve previously read about the carbon cycle. This is important, because any strategy for stabilization of atmospheric CO₂ is completely dependent on our understanding of the carbon cycle.

What is the carbon cycle?  Carbon exists in various forms and reservoirs.  Many of these exchange carbon with each other at different rates.  The major reservoirs are, in no particular order, carbon dioxide in the atmosphere, carbonate rocks (limestone and dolomite, e.g.) dissolved inorganic carbon in surface water and the ocean (carbonic acid, bicarbonate and carbonate ions and suspended calcium carbonate), fossil fuels (various forms of coal, oil shale, tar sands, petroleum and natural gas) and organic carbon in the biosphere on land and in the ocean.  The seasonal, annual and centennial scale exchanges between these reservoirs are components of the carbon cycle that are of interest. Millennial scale or longer exchange times are too slow to be of interest for modeling the effect of fossil fuel combustion and land use changes on atmospheric carbon dioxide and its potential effect on climate.

IPCC Carbon Cycle Model

The IPCC AR4 WG1 report uses the following model of the carbon cycle to determine the effect of carbon emissions on atmospheric CO2 concentration:

The CO₂ response function used in this report is based on the revised version of the Bern Carbon cycle model used in Chapter 10 of this report (Bern2.5CC; Joos et al. 2001) using a background CO₂ concentration value of 378 ppm. The decay of a pulse of CO₂ with time t is given by

ao = sum[i = 1,2,3] (ai e -t/Tau i)  — Sorry for the nomenclature, my screen capture isn’t set up since the hd kicked the bucket – Jeff

Where a₀ = 0.217, a₁ = 0.259, a₂ = 0.338, a₃ = 0.186, τ₁ = 172.9 years, τ₂ = 18.51 years, and τ₃ = 1.186 years.

So what does that mean?  There are three reservoirs other than the atmosphere where carbon can go.  They equilibrate at different rates.  At equilibrium after an addition of CO₂, 21.7% will remain in the atmosphere, 25.9% will be in a reservoir with a time constant of 172.9 years, probably intermediate depth in the ocean, 33.8% will be in a reservoir with a time constant of 18.51 years, probably the near surface ocean, and 18.6% will be in a reservoir with a time constant of 1.186 years, probably the ocean surface layer. I’ll need to introduce the concept of a continuous flow stirred reactor to show how the math applies.

Take two tanks each containing 100 kg of water.  Each tank is well stirred.  If I replace 1 kg of water with 1 kg of sodium chloride in tank 1, then the concentration of salt will be 1% in tank 1 and 0% in tank 2.  Now I take 10 kg of solution from each tank and place it in the other.  The concentration of NaCl in tank 1 is 0.9% and 0.1% in tank 2. If I keep doing this once an hour, the concentration with time looks like this:

If I subtract the equilibrium concentration of 0.5 from the tank 1 results, I can fit the data with an exponential function.  That makes the equation describing the concentration with time in tank 1 equal to:

Or, to put the exponent in the same form as the IPCC equation, it would equal –t/4.482.  So the equation constant represents the ratio of the volumes of the tanks and the time constant is proportional to the residence time, a function of the flow rate between the two tanks and the volumes of the tanks. If I continuously pump between the tanks at a flow rate of 10kg/hour, the time constant becomes equal to half the time necessary to completely replace the tank volume or 5 hours.

So the IPCC model has four tanks and the flows are only between tank 1, the atmosphere, and each of the other tanks with no flow between the other tanks.  Does it work?  Well, it turns out that there is test data available.  The Law Dome ice core has annual CO₂ concentration from 1832 to 1978 and Mauna Loa CO₂ data runs from 1959 to the present.  Total fossil fuel and cement production and land use CO₂ emissions data are also available from 1850 to 2005.  I can treat each year as a separate pulse of CO2 and let it decay over time then sum the total concentration for each year.  I can use Trenberth’s conversion factor for Mt carbon to ppmv CO₂ of 2130 MtC/ppmv.  Then I can plot the Law Dome and Mauna Loa data on the same graph using a secondary y axis to correct for the initial offset.  When I do that I get this graph:

I was quite surprised that the fit from 1850 to about 1940 was so good.  But then see what happens.  To show in more detail, I can subtract offset adjusted model concentration from the measured concentration here;

Conclusion

Obviously, CO₂ is going away compared to the model.  There was a major shift about 1937 resulting in the loss of quite a lot of CO2 from the system, on the order of 100,000 MtC when you correct for the transfer to the other reservoirs.  The loss has not increased since about 1993, but who knows how much longer that will continue? It’s possible that carbon emissions have been overestimated for the 50 years from 1940 to 1990, but what does that say about our ability to forecast emissions for the next 100 years? Combine an overestimate of how much emissions end up in the atmosphere with extremely optimistic scenarios of fossil fuel consumption for the next 100 years and it becomes very hard to place much confidence in the forecasts whatever one thinks about the accuracy of model predictions of mean temperature.