Guest essay by Joe Born
Is the Bern Model non-physical? Maybe, but not because it requires the atmosphere to partition its carbon content non-physically.
A Bern Model for the response of atmospheric carbon dioxide concentration to anthropogenic emissions is arrived at by adopting the values of , (and maybe ) that make the best fit of the following equation to the historical record:
The “Bern TAR” parameters thus adopted state that the carbon-dioxide-concentration increment initially caused by a slug of additional carbon dioxide will decay as follows:
where the ‘s are coefficients that sum to unity, the ‘s are explicit time constants of 2.57, 18.0, and 171 years, and a time constant of infinity is implicitly assigned to : of the concentration increase persists forever.
There are a lot of valid reasons not to like what that equation says, the principal one, in my view, being that the emissions and concentration record we have is too short to enable us to infer such a long time constant. What may be less valid is what I’ll call the “partitioning” version of the argument that the Bern model is non-physical.
That version of the argument was the subject of “The Bern Model Puzzle.” According to that post, the Bern Model “says that the CO2 in the air is somehow partitioned, and that the different partitions are sequestered at different rates. . . . Why don’t the fast-acting sinks just soak up the excess CO2, leaving nothing for the long-term, slow-acting sinks? I mean, if some 13% of the CO2 excess is supposed to hang around in the atmosphere for 371.3 years . . . how do the fast-acting sinks know to not just absorb it before the slow sinks get to it?” (The 371.3 years came from another parameter set suggested for the Bern Model.)
The comments that followed that post included several by Robert Brown in which he advanced other grounds for considering the Bern Model non-physical. As to the partitioning argument, though, one of his comments actually came tantalizingly close to refuting it. Now, it’s not clear that doing so was his intention. And, in any event, he did not really lay out how the circuit he drew (almost) answered the partitioning argument.
So this post will flesh the answer out by observing that the response defined by the “Bern TAR” parameters is simply the solution to the following equation:
But that equation describes the system that the accompanying diagram depicts. And that system does not impose partitioning of the type that the above-cited post describes.
In the depicted system, four vessels of respective fixed volumes contain respective variable quantities of an ideal gas, which they keep at a constant temperature so that the pressure in each vessel is proportional to its respective value of . The vessel on the left exchanges gas with each vessel on the right through membranes of respective permeabilities , the net rate of gas exchange with a given vessel on the right being proportional to the difference between that vessel’s pressure and the left vessel’s pressure. For the th vessel on the right, that is,
Additionally, a gas source can add gas to the first vessel at a rate , so the left vessel’s contents can be found by solving the following equation:
If appropriate selections are made for the ‘s and ‘s, then expressing the other ‘s in terms of converts that equation into the fourth-order equation above, i.e., into the system equation that the “Bern TAR” parameters dictate.
The gas represents carbon (typically as a constituent of carbon dioxide, cellulose, etc.), the first vessel represents the atmosphere, the other vessels represent other parts of the carbon cycle, the membranes represent processes such as photosynthesis, absorption, and respiration, and the stimulus represents the rate at which carbon rejoins the carbon cycle after having been lost to it for eons.
I digress here to draw attention to the fact that I’ve just moved the pea. The flow from the source does not represent all emissions, or even all anthropogenic emissions. It represents the flow only of carbon that had previously been sequestered for geological periods as, e.g., coal, and that is now being returned to the cycle of life. Thus re-defining the model’s emissions quantity finesses the objection some have made that the Bern Model requires either that processes (implausibly) distinguish between anthropogenic and natural carbon-dioxide molecules or that atmospheric carbon dioxide increase without limit.
Now, there’s a lot to criticize about the Bern Model; many of the criticisms can be found in the reader comments that followed the partitioning-argument post. Notable among those were richardscourtney’s . Also persuasive to me was Dr. Brown’s observation that the atmosphere holds too small a portion of the total carbon-cycle content for the 0.152 value assigned to the infinite-time-constant component to be correct. And much in Ferdinand Engelbeen’s oeuvre is no doubt relevant to the issue.
As the diagram shows, though, the left, atmosphere-representing vessel receives all the emissions, and it permits all of the other vessels to compete freely for its contents according to their respective membranes’ permeabilities. So what is not wrong with the model is that it requires the atmosphere to partition its contents, i.e., to withhold some of its contents from the faster processes so that the slower ones get the share that the model dictates.
- On CO2 residence times: The chicken or the egg? (wattsupwiththat.com)