Carbon Circular logic

The cornerstone of climate science must be the enhanced greenhouse effect caused by rising CO2 levels. Therefore understanding how CO2 may increase in the future is of fundamental importance, and you would assume that any Earth Systems Model(ESM) should as a minimum be able to describe past increases in CO2.  I had naively assumed that the BERN model was tuned so as to match emissions to CO2 increases, but it turns out this is not true.

Here is the result for CO2 levels if you integrate the AR4 Bern model using the historic emission data (CDIAC) and then compare it to the actual CO2 measurements from Dome-C and Moana Loa. (see last post for details)

Comparison of cumulative emissions (CDIAC) and CO2 measurements. The purple curve is AR4 Bern model. The red curve is TAR Bern model. The data are Dome-C (Epica) CO2 data (gold) and Moana Loa + global averaged CO2 data (green).

There are two clear problems. Firstly the CO2 measurements rise well above cumulative emissions from 1800 to 1950,  which is seemingly impossible. The second problem is that even after 1950, the AR4 Bern model only begins to agree with ML after 2000 and even then cannot reproduce the time trend. So what is going wrong or did I perhaps miss something?  Well I have not included Land Use ’emissions’ which has been estimated from historical records of deforestation. Deforestation reduces net carbon uptake by living biota leading to an annual ’emission’. To remedy this I have simply used the AR5 Land Use data from this R.J.Houghton’s study (CDIAC).

Here is the new result after adding in the ‘land use’ data to the emissions data, where I now also use the latest BERN multi-model result (see Joos et al. 2013) .

Calculated CO2 levels including Land Use Emissions. Curves are the latest BERN model parameters (Joos et al. 2013) which are essentially the same as the AR4 values above within this time frame. Dashed purple is the same model bit excluding Land-Use values.

This gives a better agreement with the Dome-C data up until 1950, which now also lies correctly below the cumulative emissions.  However the BERN model  significantly overestimates CO2 levels thereafter. This demonstrates that merely adding in land use emissions still does not reproduce past CO2 levels. Nor am I alone in discovering this discrepancy.

A recent discussion paper by Millar et  al.  which uses a fit to ESMs to derive a ‘Finite Amplitude Impulse Response (FAIR)  function, also fails to reproduce the actual measured CO2 levels. The AR5-IR model as described in the AR5 report does even worse.

Figure 4a) shows a reconstruction their ‘FAIR’ model and an AR5-IR model integrated using emissions data. The AR5-IR model fails miserably while their FAIR model also overshoots emissions. The actual emissions  data used are those shown in Figure 4b) while the blue curve is what hypothetical emissions would have been needed to reproduce the actual CO2 Keeling curve. Their result is essentially the same as mine. A boost in emissions is needed before 1930 to explain the dome-C measurements and the AR5 model overestimates CO2 levels.

A NASA paper recently reported on the CO2 greening effect. So did somehow the global biota switch from being a source of emissions (deforestation) to being a net sink (greening) as CO2 levels increased? Suddenly everyone thinks this was obvious all along.

Prof Corinne Le Quéré, director of the Tyndall Centre at the University of East Anglia said: “Natural vegetation is a fantastic help in slowing down climate change by absorbing about a quarter of our carbon emissions from burning fossil fuels.

Yet it seems clear to me that ESMs did not take sufficient account of any fertilisation effect. Nor did AR5 report any global uptake from CO2 fertilised growth in plants either. It merely refers to possible regrowth from past clearances in specific regions.

The second major source of anthropogenic CO2 emissions to the atmosphere is caused by changes in land use (mainly deforestation), which causes globally a net reduction in land carbon storage, although recovery from past land use change can cause a net gain in land carbon storage in some regions.

Global net CO2 emissions from land use change are estimated at 1.4, 1.5 and 1.1 PgC yr–1 for the 1980s, 1990s and 2000s, respectively, by the bookkeeping method of Houghton et al. (2012)

This makes it clear that any fertilisation effect was also ignored by ESMs. This likely means that IPCC projections of future CO2 levels based on the Bern model are too high.

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Bern Model when emissions stabilised at 2013 levels

The Bern model has been used by IPCC for emission scenarios since SAR in 1995. We will now use it to extrapolate CO2 levels 500 years into the future with annual emissions fixed at 2013 levels (~10 GT C/year). All values are expressed as an equivalent atmospheric CO2 concentration in ppm for convenience.

To do this we integrate emissions over time and and calculate atmospheric CO2 concentrations.

CO_2(t) = CO_2(1750) + fac \times \int_{1750}^{t} Em(t') \times ( a_0 + \sum_{1}^{3} a_i e^{-\frac{t-t'}{\tau_i}} ) dt'  

where fac converts GTC to ppm, Em = emissions in year t’ and the a’s and are \tau's are the parameters of the Bern model. CO_2(1750) = 280ppm . For the integration I use this dataset for the emissions data.

The Bern model simply describes the time decay of an annual pulse of CO2 added to the atmosphere. It is parameterised as follows.

\Delta C = a_0 + \sum_{1}^{3} a_i e^{-\frac{t}{\tau_i}}

where for AR4:  a_0 = 0.217,  a_1 = 0.259,  a_2 = 0.338,  a_3 = 0.186,  \tau_1 = 172.9 y ,  \tau_2 = 18.51 y , \tau_3 = 1.186 y

and for TAR:   a_0 = 0.152,  a_1 = 0.253,  a_2 = 0.279,  a_3 = 0.316

Here are  the results.

CO2 levels in the atmosphere following a stabilisation at 2013 emission rates held constant for 500 years.

Levels do not reach an equilibrium value in the Bern model because a fixed fraction (a0) in any year are assumed to remain for ever in the atmosphere. However even in this case levels rise (just) to ~1270 ppm with AR4 parameters and ~950 ppm using TAR values. The airborne fraction shows how it falls asymptotically to the fixed retention rate.

Airborne fraction for the Bern model 500 years into the future for fixed 2013 emissions.

We can also ask the question as to how well the Bern model describes the measured CO2 levels between 1950 to 2016 based only on the emissions data. Here is that comparison.

Detail comparison. The dark green trace is the Moana Loa CO2 measurements .

The bern model agrees with the current value of CO2 (400ppm), but it does not give a good description of the trend. In fact the actual CO2 growth is slower than that of the model. This supports the hypothesis that a0 is actually very small and that CO2 levels will approach stability much faster.

Let’s consider two estimates of climate sensitivity (ECS) a median value of 2.5C (preferred by Gavin Schmidt) and a low value of 1.5C (preferred by say Nic Lewis).

ECS Model Version Net Warming in 2620
2.5C AR4 5.5C
2.5C TAR 4.2C
1.5C AR4 3.2C
1.5C TAR 2.5C

When you consider that we are assuming annual emissions held constant at ~10 GtC/y for 500y into the future, then these final result are not really that scary at all.

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A realistic CO2 stabilisation scenario

This is a guest post by “Frank” based on his comment to  Stabilising Climate. He argues that CO2 levels will stabilise at ~520 ppm, which agrees with an estimate given by Ferdinand  Engelbeen here quoted below:

My rough estimate is that with the current emissions twice the current sink rate at ~110 ppmv above steady state, one need ~220 ppmv above steady state to get rid of the full ~4.3 ppmv/year human emissions. That is a level of 510 ppmv, far above the 440 ppmv of Clive… The observed e-fold decay rate of the extra CO2 in the atmosphere is around 51 years, surprisingly linear over the past 57 years. No reduction in sink rate to see

Now here is Frank’s argument

Man is currently emitting enough CO2 to increase atmospheric CO2 by 4 ppm/yr, but we only observe an increase in atmospheric CO2 of 2 ppm/yr. Presumably that means that 400 ppm of CO2 is enough to drive 2 ppm/yr from the atmosphere. My intuition says that if we cut emissions in half, sinks would continue to take up 2 ppm/yr (until sinks begin to “saturate”) and atmospheric CO2 would stabilize at the current level. On the other hand, you say that a 1% increase in CO2 to 404 ppm is enough to double the rate at which CO2 disappears into sinks from 2 to 4 ppm/yr.  Since this is a blog, I simply talk about emissions and uptake in units of ppm/yr rather than Gtons/year – it is much more intuitive and the math is far simpler.

I think maybe this is the correct mathematical formulation :

\frac{d(CO2_{atm})}{dt} = -k_{un}(CO2_{atm}) + k_{en1}(CO2_{sink1}) + k_{en2}(CO2_{sink2}) + k_{ea}

\frac{d(CO2_{atm})}{dt}  = rate of change in CO2_{atm}
k_{un} = rate constant for Uptake of CO2 by all Natural sinks.
k_{en1} = rate constant for Emission of CO2 by Sink #1 by Natural processes.
CO2_{sink1} = concentration of CO2 in Sink #1. Could be DIC in ocean.
k_{en2} = rate constant for Emission of CO2 by Sink #2 by Natural processes.
CO2_{sink2} = conc. of CO2 in Sink #2. Could be organic material in soil in CO2 equiv. Or plant matter growing above ground. Some of this is emitted as CH4 and then oxidized, … Other sinks and rate constants.
k_{ea} = rate of Emission of CO2 by Anthropogenic mechanisms.

During the 10 millennia before the Industrial Revolution, \frac{d(CO2_{atm})}{dt} and k_{ea} were both near zero and a steady state relationship existed.

k_{un}(CO2_{atm} = 280 ppm) = k_{en1}(CO2_{sink1}) + k_{en2}(CO2_{sink2})

If we assume that the size of the sinks is much larger than the amount of CO2 they have taken up so far,  and that their rate constants are independent of global warming, then the second and third terms on the right hand side haven’t changed. For the largest sink that is the deep ocean, we know that the MOC takes about a millennia to overturn the ocean. So the deep ocean sink isn’t going to saturate in the near future. The mixed layer of the ocean is rapidly mixed by wind, so the CO2 content of the mixed layer is always in equilibrium with the atmosphere and that equilibrium is only slightly temperature sensitive. It doesn’t saturate either. The land sinks are a little trickier, but let’s make the assumption their rate constants and capacity haven’t yet changed appreciably. So we can substitute:

\frac{d(CO2_{atm})}{dt} = -k_{un}(CO2_{atm}) + k_{un}(CO2_{atm} = 280 ppm) + k_{ea}


\frac{d(CO2_{atm})}{dt} = 2 ppm/yr
CO2_{atm} = 400 ppm
k_{ea} = 4 ppm/yr


2 ppm/yr = -k_{un}[120 ppm] + 4 ppm/yr
k_{un} = 1/60 yr^{-1}

To double-check for consistency, go back to 1960 when (IIRC)

\frac{d(CO2_{atm})}{dt} = 1 ppm/yr
CO2_{atm} = 330 ppm
k_{ea} = 2 ppm/yr

1 ppm/yr =-k_{un}[50 ppm] + 2 ppm/yr
k_{un} = 1/50 yr^{-1}

I really should look up the 1960 values and not trust my memory. However, no sign of saturation here either.

So what happens if we continue to emit a constant (not growing) 4 ppm/yr, CO2_{atm} = 404 ppm and nothing else changes:

\frac{d(CO2_{atm})}{dt} = -\frac{1}{60} \times [124 ppm] + 4 ppm = +1.93 ppm/yr

Therefore CO2_{atm} needs to rise to 520 ppm (280+240) for \frac{d(CO2_{atm})}{dt} to be zero and atmospheric CO2 to stabilize when we are emitting the equivalent of 4 ppm/yr.

And my intuitive answer that we need to cut back to 2 ppm/yr to stabilize near 400 ppm (for as long as the sinks don’t saturate) agrees with this mathematics.

Are sinks saturating?

Reservoirs of carbon (in GtC) in the ocean (blue labels), in biomass in the sea and on land (tan and green labels), in the atmosphere (light blue label) and in anthropogenic emissions. Fluxes of Carbon between reservoirs are depicted by the arrows, the numbers represent GtC. (From: IPCC)

We’ve emitted 240 ppm of CO2, 120 ppm is in the atmosphere, 60 ppm-equivalents is in the ocean, and 60 ppm-equivalents on land. This Figure is using units of GtC and 400 ppm = 750 GtC. The land biomass reservoir is about 2000 ppm-eq and the deep ocean reservoir is about 20,000 ppm-eq. The increased CO2 stored in these reservoirs since 1750 is trivial compared with their size, so there isn’t an obvious reason why emission from the reservoirs should have increased already or will increase in the future.

In my main equation above, I should include a term for the increase in photosynthesis (primary productivity) with rising CO2. The incorporation of CO2 into organic material is the rate limiting step, so it could also have increased by a factor of 400/280 – at least in areas where water and other nutrients (N, P, K, and micronutrients) are not limiting.

CB comment:  No-one is suggesting that it is a good idea to keep CO2 emissions at current levels for centuries to come, but stabilising emissions now is a more realistic goal than reducing them to zero, and limits atmospheric CO2 levels. This gives more time to develop a far more rational future energy and transport policy.

Posted in AGW, Climate Change, climate science | Tagged , | 27 Comments