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.

where fac converts GTC to ppm, Em = emissions in year t’ and the a’s and are are the parameters of the Bern model. . 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.

where for AR4:

and for TAR:

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.

*Related*

You have two green lines that seem to overlap–Mauna Loa CO2 measurements and then something called Av-Measurements (undefined). Is that the average of all other CO2 measuring locations?

Yes it is an average of other spatialy distributes measurements since 1980. It changes very little.

Clive, this is very informative. It looks from your chart that Bern Model over-estimates the trend for CO2 accumulation (or underestimates the CO2 natural sinks). I would also be interested to see a chart of a simulation of emissions diminishing over the next 83 years to zero, to reflect an assumption that we would be an entirely nuclear fusion economy by then, able to create synthetic carbon fuels cheaply be then by reduction of CO2.

On the quoting of Gavin’s ECS, I don’t dispute it but I would love to have a neat reference if you have, and also for other leading IPCC figures.

It would be easy to put in a gradual reduction into the emissions trend. However, the physical assumption built in to the Bern model is that 20% of emissions from any year remain in the atmosphere forever. This cannot make physical sense. The only thing that does make sense is that the balance of PCO2 is reached first with the surface layer which then ‘diffuses’ down to the deep ocean over a longer time scale, allowing more absorption. Eventually all the ocean is ‘balanced’ so supposedly the rest has to wait for geological burial, but why 20% and not say 2%? Why would it even be constant?

Gavin quote is from http://www.carbonbrief.org/the-carbon-brief-interview-dr-gavin-schmidt

OK he says 2.5C-3.0C, but I seem to remember somewhere else when he was forced to give a preference and he said 2.5C

On Twitter

?@clivehbest

@ClimateOfGavin @Balinteractive When you say climate sensitivity is probably around 2.5-3C

I assume you mean ECS. Correct?6:14 PM – 16 Oct 2015

Gavin Schmidt ?@ClimateOfGavin 16 Oct 2015

@clivehbest

Yes.@Balinteractive1 reply 0 retweets 0 likes

None of this has any relevance because the basic premise that carbon dioxide has any affect on climate is wrong. Solar energy controls climate.

Hi Clive,

The form of the atm decay function is an interesting question w/ uncertainty. But I’m wondering where you got your model parameters because they don’t match the ones given by Joos et al 2013, from which I got mine. Thanks.

I programmed up the Joos et al. 2013 values and although they are very different the end result is more or less the same as those quoted and displayed above in AR4 (labelled here as AR5)

Thanks for so doing Clive. I have to believe the high similarity of result with very different parameters, means simply a difference in time step between the two.

I’m thinking about this issue a lot and have a post partially written but I’m moving at a glacial pace right now. Based on some fairly simple math, I cannot in any way see how they come up with such long, slow relaxation rates to a high asymptote. Strongly skeptical on that concept. And this in turn has implications for estimated GWPs (global warming potentials), via direct effects on time-integrated RF.

Your statement in the other post, to a commenter, regarding the idea that continually increasing emissions is masking detection of faster relaxation times, is both interesting and quite important. I’m thinking a lot about that.

This is quite a big deal.