CO2 Party: Having Fun with Probabilities

CO2 Party: Having Fun with Probabilities

By Clyde Spencer | Dec. 16, 2021


Imagine that someone decides to throw a theme party and the theme they choose is ‘blue.’  They invite 10 of their friends and ask each of them to bring a plastic baggie filled with blue M&Ms™ candies.  When the guests arrive, they empty their baggies into an empty punchbowl.  The host(ess) places the filled bowl on the hors d’oeuvres table.  Assuming that each of the guests brings on average about 100 pieces, there will be about 1,000 pieces total.  Throughout the evening, the guests partake sparingly of the contents of the punchbowl (the host(ess) abstains.).  At the end of the evening, there are still some candy pieces left.  As they are leaving, one of the guests claims them, boldly stating that there are about half as many candy pieces as he brought, and they must therefore be the same ones he brought, despite having brought noticeably fewer pieces of candy than the others had!  Other than being a bit boorish, what can we conclude about the claim?

Strictly speaking, we are dealing with a situation of sampling without replacement, which means that the probability of drawing a piece of candy contributed by any person varies over time, depending on how many of a particular person’s contribution remains after withdrawals.  Unfortunately, we don’t have that information.  I’m assuming, for the sake of illustration, that approximately equal numbers of candy from all the guests are drawn, and that the total number of pieces of candy is large enough that, at least initially, there is a negligible change in the ratio of the pieces drawn to the number remaining.  Therefore, the probability remains approximately constant until we get below 10 pieces.  Thus, the following reliance on the initial probability.  As is the practice in climatology, I’m going to ignore probability uncertainties and their propagation.  I’ll just work with orders of magnitude.

If there were only one piece of candy left, we could trivially conclude that there was 100% probability that one person brought it.  However, who?  Probably the first person to arrive and put their candy in the bowl if the contents hadn’t been mixed.  FILO – First In, Last Out!    Alternatively, more probably, the person who brought the most candy, if the contents are well mixed.  However, in the absence of information on the quantity and order of the candy placed in the bowl, the best we can probably do, assuming that everyone brought approximately the same number of pieces of candy, is to say that the single remaining piece has about a 1:10 chance of belonging to a particular person, or 0.1.  Although, that doesn’t allow us to determine who that person is.

Things get a little more interesting and complicated if there are two pieces of candy left over.  What is the chance that the same person brought both pieces?  The probability of a sequence of events is the product of the probabilities of each event.  That probability is about 0.1 x 0.1 or 0.01, in the well-mixed case, which I will assume.  What if there are five pieces left?  The probability that the same person brought all five remaining pieces would be about 0.1 raised to the 5th power, or 0.15 ≈ 10-5.  It should be obvious that attribution of source rapidly becomes uncertain as the number of sources increases and the number of events (pieces of candy) increases!  Therefore, it becomes very unlikely that the same person brought all of the remaining pieces.  That is, having a large number of pieces of candy left over, all from the same person is highly improbable.  However, the probability increases to 1 as a limit as the number of pieces of candy declines to 1.


In the above story, the punchbowl represents the tropospheric atmosphere, the contributed blue M&Ms the annual flux of well-mixed CO2 that is added over the Winter, and the candy consumed represents the annual flux of CO2 that is captured by the global sinks, principally during the Summer.  The number of pieces of candy remaining at the end of the party represents the annual net increase in CO2.  It is claimed commonly that, because the atmospheric concentration of CO2 is increasing annually by an amount that is almost half the estimated anthropogenic emissions, humans are solely responsible for the increase in atmospheric CO2, and ergo, eliminating anthropogenic emissions will stop the rise of CO2 and therefore stop the rise in temperature of the globe.

One problem with the assumption that only anthropogenic emissions are responsible for the annual increase in CO2 is that there is no empirical evidence for it.  The decline in anthropogenic emissions during the height of the COVID pandemic did not result in any measureable decline in the total increase during 2020, or rate of increase for any of the months; nor was the decline faster than typical.  I have discussed this in detail here:

Summarizing the above linked article, the atmospheric CO2 concentration varies seasonally.  It increases about 8 PPMv from Oct thru May, and decreases about 6 PPMv from June thru Sept.  During the ramp-up phase, Fall thru early-Spring, photosynthesis is significantly reduced and the net change is an increase in atmospheric CO2 concentration.  However, during April of 2020, there was a pandemic-induced decline of about 18% in anthropogenic CO2, but there was no observable change in the rate of increase; the curve essentially looked like the previous year.  Similarly, the maximum concentration reached in May was virtually the same as in 2018-2019, despite there being reduced estimated anthropogenic CO2 emissions, December 2019 through May 2020.

The anthropogenic sources of CO2, not all of which are from burning fossil fuels, only amount to about 4% of the total CO2 flux in the Carbon Cycle, which strongly suggests that the small flux of anthropogenic CO2 is dwarfed by the biogenic sources and outgassing from warming water, leading to a negligible residual anthropogenic accumulation in the atmosphere.

All CO2 is partitioned into the various sinks (air, water, terrestrial plants, phytoplankton) in proportion to the fractional abundance compared to the annual total.  The sinks cannot tell the difference between CO2 sourced from fossil fuels, plant respiration, or bacterial decomposition!  That is, if all fossil fuel emissions were to magically cease tomorrow, we could only expect to see <4% decline in the rate of atmospheric CO2 concentration growth, not the 50% we are being told to expect.

The problem is that sources and sinks are more sensitive to the abundance of CO2 (partial pressure) than other differences such as the atomic weight of the CO2 molecules.  Therefore, the sources can’t significantly differentiate between anthropogenic and natural sources, such as biogenic CO2 or ocean outgassing.  The same is true for sinks, with the notable exception of photosynthetic organisms showing a slight preference for light CO2 molecules with a 12C isotope.  That is the point of the little story above about the M&Ms.  That is, if the person claiming the remaining pieces of candy had not brought any, there would still probably be some candy remaining, although it obviously could not have been his.

Another way of looking at this issue is that, for a first-order approximation ignoring isotopic fractionation, the sinks should extract CO2 out of the atmosphere in direct proportion to the relative abundance of the source CO2.  That is, if there is a net annual gain of 2 or 3 PPM, almost all of that has to be from the sources with the greatest abundance – oceanic out-gassing and biogenic respiration.  The same argument about the trivial contribution from volcanic activity applies equally to anthropogenic emissions.

Most of the claimed supporting evidence for anthropogenic CO2 concentrating in the atmosphere is based on changes in the isotopic carbon proportions.  The argument is that fossil fuels have a small deficit of 13C and the measured increase in the relative proportion of atmospheric 12C must therefore be from CO2 derived from fossil fuels.  The situation is more complex than suggested because recent work (Kieft, et al., 2021) has shown that bacterial recycling of dissolved organic matter in the oceans may concentrate the 13C isotope!

During nighttime, plants respire CO2.  Dormant deciduous trees still respire (during Winter) through their roots.  However, evergreen trees in boreal forests respire more because they retain their needles.  I would expect this respiration, which contributes to the Winter CO2 ramp-up to be deficient in 13C.

Another flaw in the isotope defense is that there should be a preference for light (12C-rich) CO2 outgassing from the ocean surface because it takes less energy for wind to strip it out than for the heavier molecules.  I’m unaware of anyone having taken this into consideration when defending the claim of the increase in atmospheric CO2 being the result of anthropogenic emissions, despite some early work having been done (Doctor, et al., 2008) with freshwater.  Additionally, Mayorga et al. (2012) show that isotopic fractionation occurs between the dissolved carbon species carbonic acid, aqueous bicarbonate, and aqueous carbonate, during conversion between species, with pH change, as well as with outgassing.  Earlier work by Wanninkhof (1985) left some questions unanswered, but stated:

“A box model of Keeling et al. (1980) shows a difference in δ13C change in the atmosphere from 1956 to 1978 of 0.15 ‰ depending on whether an air-seawater fractionation constant of -14 ‰ or 0 ‰ is used.  This is quite significant if we consider that the total δ13C change in the atmosphere for the past 100 years is about -I ‰, based on tree ring data (Peng et al., 1983).”


Since the launching, in late-2014, of the Orbiting Carbon Observatory-2 (OCO-2) satellite, I have seen many CO2 maps.  I was unable to find most of them with a general online search.  They were not available at the NASA JPL OCO-2 website.  The entire archive apparently has been reprocessed, but all that I was able to find was 2015 through 2017 data.  At least one video was deleted (the link is not functioning) from the NASA JPL OCO-2 website.  The recent maps are not as user friendly as the original graphics released to the public.  In searching for suitable OCO-2 CO2 maps, I was impressed by two things: 1) How difficult it was to find previously published maps, and 2) How much variation there was in the few available maps.

Despite being characterized as “well-mixed,” within the limits of quantitative resolution, CO2 varies considerably in concentration, location, and with the seasons.  The earliest CO2 map from the OCO-2 satellite is probably the most useful for this discussion because it shows the distribution of concentrations for a 5-week period during the beginning (low point) of the seasonal ramp-up phase for the northern hemisphere (NH).

Figure 1a (below) is the first release of OCO-2 data at the 2014 American Geophysical Union meeting.  It appears that the major sources are on land, such as the Amazon Basin and southern Africa, with secondary sources from outgassing in the oceans in an Equatorial belt.  These are not regions of either high population density or concentrated industrial activity.

Following that up with another map, Figure 1b, made with data from about two months later, shows how much the location of the major sources changed in just a month in the early NH ramp-up phase.  None of the red and little of the yellow that is shown is from cars or factories.  Clearly, natural biogenic sources associated with decaying detritus lying on the ground, and evergreen tree respiration, particularly in the boreal forests of North America and Siberia, dominate the Northern Hemisphere sources.  The outgassing from the tropical oceans is gone, perhaps because it is early-Winter and the surfaces waters have cooled.  It appears that there is still a band of northerly CO2 source from the ocean; however, it may be the result of dead, decomposing phytoplankton still near the surface.


The curve for the 2014-2015 CO2 ramp-up phase (See Figure 2, below.) is typical for the last 30-years, albeit the maximum in May is lower than in recent years.  However, the following year was an El Niño year and the May high was typical of recent years.  This suggests temperature controlling the CO2 concentration.

Note that the deviations from the linear regression lines recur in most years and are not just random variations in interannual variance.

Fundamentally, it appears that the increase in CO2, as exhibited during the Fall-Spring ramp-up phase, is not being matched by the drawdown phase in Summer, despite the slope of the Summer curve being steeper.

Figure 2.  CO2 concentrations 2014-2015 ramp-up and drawdown phases.

The months marked in blue (1 & 3) correspond to the two maps in Fig. 1a and 1b.

SUMMARY TIME (and the living is easy)

The major sources of CO2 are not spatially associated with high population densities or industrial activity during the seasonal ramp-up phase, with the possible exception of China.

It is improbable that more than a small fraction of the annual anthropogenic emissions remain in the atmosphere because its proportion of total source annual-flux is <4%.  The stated fact that the annual increase in atmospheric concentration of CO2 is about one-half the anthropogenic emissions is probably a spurious correlation.

The accounting for the change in atmospheric CO2 isotopic composition resulting from fossil fuel emissions is not rigorous for all the potential sources of isotopic fractionation.

An alternative interpretation for the current paradigm is that, against a background of relatively constant anthropogenic emissions, the warming Earth forces an increase in ocean out-gassing and biogenic emissions during the seasonal CO2 ramp-up phase.  During the drawdown phase, the warming high-latitude waters are less effective at capturing the CO2 in the atmosphere.  Also, during the drawdown phase, the increased CO2 in the atmosphere results in increased growth of vegetation and photosynthetic plankton; however, the increase is only sufficient to capture an amount of CO2 that is equivalent to about half of the annual anthropogenic emissions.  Therefore, in the absence of anthropogenic emissions, one might expect the growth in atmospheric CO2 to be 96% of the current total annual CO2 flux.  The average annual net growth in atmospheric CO2 is about 1.8 PPM over the last 30 years.  Therefore, one could expect that in the absence of anthropogenic CO2, the annual increase might be about 1.7 PPM.  However, because fossil fuels only represent about 95% of anthropogenic emissions, and it is impractical to stop making cement and quit using CO2 as an industrial feedstock, the net annual gain would be somewhat greater than 1.7 PPM.  Thus, even draconian emission reductions of anthropogenic CO2 cannot be expected to have more than negligible effect!

It is a common alarmist refrain that when temperatures go down, it is weather; however, when temperatures go up, they call it climate.  There is a similar situation with atmospheric CO2.  When atmospheric concentrations go up, it is claimed to be solely the result of increasing anthropogenic emissions.  When anthropogenic emissions go down, we are told that natural variability masks the expected decrease.


Brandon Kieft, Zhou Li, Samuel Bryson, Robert L. Hettich, Chongle Pan, Xavier Mayali, Ryan S. Mueller (2021).  Phytoplankton exudates and lysates support distinct microbial consortia with specialized metabolic and ecophysiological traits. Proceedings of the National Academy of Sciences Oct 2021, 118 (41) e2101178118; DOI: 10.1073/pnas.2101178118

Doctor, D. H., Kendall, C., Sebestyen, S. D., Shanley, J. B., Ohte, N., & Boyer, E. W. (2008). Carbon isotope fractionation of dissolved inorganic carbon (DIC) due to outgassing of carbon dioxide from a headwater stream. Hydrological Processes, 22(14), 2410-2423.

Mayorga, E., A.K. Aufdenkampe, C.A. Masiello, A.V. Krusche, J.I. Hedges, P.D. Quay, J.E. Richey, and T.A. Brown. (2012). LBA-ECO CD-06 Isotopic Composition of Carbon Fractions, Amazon Basin River Water. Data set. Available on-line [ ] from Oak Ridge National Laboratory Distributed Active Archive Center, Oak Ridge, Tennessee, U.S.A.

Wanninkhof, Rik (1985) Kinetic fractionation of the carbon isotopes 13C and 12C
during transfer of CO2 from air to seawater, Tellus B: Chemical and Physical Meteorology, 37:3,
128-135, DOI: 10.3402/tellusb.v37i3.15008

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