Guest Post by Willis Eschenbach

A decade ago I wrote a post entitled “The Bern Model Puzzle”. It related to the following question.

Suppose we have a relatively steady-state condition, where the CO2 level in the atmosphere is neither rising or falling. Something like the situation around the year 1400 in the data below.

Figure 1. Historical airborne CO2 levels 1000AD to the present, from 10 ice cores and since 1959, from the Mauna Loa Observatory measurements (orange). Units are parts per million by volume (ppmv) of the atmosphere.

Now, suppose during that time, a volcano blows its top and dumps what we used to call a “metric buttload” of CO₂ into the atmosphere. Over time, that pulse of CO₂ will be absorbed by a variety of land and ocean sinks, and the status quo ante of atmospheric CO₂ will be restored to the level it was before the eruption.

The “Bern Model” is a model used by the IPCC and various climate models. It purports to calculate how long it takes that pulse of CO₂ to be reabsorbed by the natural sinks. And that’s where things get curious.

First off, the Bern Model says that 15.2% of that pulse of CO₂ will stay airborne forever. Not 15% of the pulse, mind you … 15.2%.

I have never found anyone who can explain this to me. If this were true, it seems to me that every volcanic eruption would lead to a new and higher permanent level of airborne CO₂ … but as you can see from Figure 1, that simply hasn’t happened.

For further evidence that the first claim of the Bern model is wrong, consider the annual swing of CO₂ levels. From a low point around October to a high point around May of each year, there is a short sharp natural pulse of CO₂ that leads to an increase in CO₂ levels of about 6 parts per million by volume (ppmv). And this is matched by an equal sequestration of CO₂ in natural sinks such that by the following October the previous CO₂ level is restored. If that were not the case, CO₂ levels would have been increasing every year since forever.

And during that same seven month period, at present we emit a pulse containing enough CO₂ to result in an increase in CO₂ levels of about 1.3 ppmv.

The Bern Model says that 15.2% of the 1.3 ppmv anthropogenic CO₂ pulse will stay in the air forever … but the ~ 6 ppmv pulse is gone very quickly. So how does nature know the difference?

But that’s just the start of the oddity. It gets more curious. The Bern Model says that :

• 25.3% of the CO₂ pulse decays back to the previous steady-state condition at a rate of 0.58% per year
• another 27.9% of the pulse decays at 5.4% per year, and
• a final 31.6% of the pulse decays back to the steady-state condition at 32.2% per year

This leads me to the same problem. How does nature know the difference? How is the CO₂ partitioned in nature? What prevents the CO₂ that’s still airborne from being sequestered by the fast-acting CO₂ sinks?

There is, however, a more fundamental problem—the Bern Model simply doesn’t do a good job at representing reality. We have reasonably good information on CO₂ emissions since 1850, available from Our World In Data. And we have reasonably good information on airborne CO₂ concentrations since 1850 from ice cores and Mauna Loa, as shown in Figure 2.

Figure 1. Historical airborne CO₂ levels from 1850AD to the present, from 10 ice cores and since 1959, from the Mauna Loa Observatory measurements (orange). Units are parts per million by volume (ppmv) of the atmosphere.

So I thought I’d take a look at the Bern Model, to see how well it could predict the airborne CO₂ since 1850 from the emissions since 1850. The equation for the calculation is in the UNFCCC paper “Parameters for tuning a simple carbon cycle model“, and is also in the endnotes … bad news.

Figure 3. Actual atmospheric CO2 values, and values according to the Bern Model

No bueno … the fact that the Bern Model results are so much smaller indicates that it is incorrectly pushing much of the effect far out into the future.

So, is there a better way? Well, yes. The better way is to use the standard lagging formula:

CO2(t+1) = CO2(t)+ λ ∆E(t) * (1- exp( -1 / τ ) + CO2(t) exp( -1 / τ )

where:

• t = time
• E(t) = emissions at time t
• CO₂(t) = CO2 concentration at time t
• λ = .47 (converts carbon emissions to ppmv)
• ∆ = difference from the previous value, so for example ∆CO₂(t) = CO₂(t) – CO₂(t-1)
• τ = tau, the time constant for the decay

Using this formula, I find the time constant tau to be ~49 years. Here’s the result of that calculation.

Figure 4. Actual atmospheric CO₂ values, and values according to a standard lagging model

This puts the halflife of a pulse of CO₂ into the atmosphere at about 34 years …

Those are my questions and observations about the Bern Model. I’ve put the calculations and data into a spreadsheet here.

Now I need to go climb on the roof and pressure-wash the cedar-shingled walls in preparation for spraying FlameStop on them … dry times in California.

My very best wishes to all, comments and questions welcome.

w.

Of Course: As is my wont, I ask that you quote the exact words you are discussing. That way, nobody’s words get misconstrued. Well, fewer peoples’ words, at least.

The Equation: As promised …

Late News: Well, I’d just finished pressure-washing the upper part of the house when my pressure washer died … and while I know you may find this hard to believe, at that point I actually said very bad words …

Looks like Dr. W. is gonna have to engage in a forensic autopsy, to see if I can perform the Lazarus trick on the !@#$%^& pressure washer.

But not today … enough. And at least the pressure washing of the upper part of the house is done, done, done.