-- by Horatio Algeranon
The upward climb
Of CO2
Has now been studied
Through and through.
The climate scientists
Are no dopes.
They track the rise
With "isotopes"
Of carbon atoms
In CO2
C13 and C12
And C14 too!
The "isotope ratio"
Is the key.
Its change over time
Is plain to see
And shows
That burning fossil fuel
Has caused the climb
As a general rule.
The claim (or implication) that "CO2 carbon isotope analysis* is not evidence in support of the argument that human fossil-fuel burning is behind the rise in atmospheric carbon dioxide concentration" has become quite popular in certain circles (and quite circular in popular circles).
*specifically, the observed decrease over time in the "carbon isotope ratio" (ratio of 13CO2 over 12CO2) -- ie, decrease in atmospheric content of 13CO2 (carbon dioxide containing the carbon 13 isotope) relative to content of 12CO2(carbon dioxide containing the carbon 12 isotope). Note: Both atmospheric 12CO2 and 13CO2 concentration have been increasing over (recent) time, but the 13CO2/12CO2 ratio has been decreasing, as shown on the graph below.
It is probably fairly safe to say that the vast majority of those who make the above claim have absolutely no clue what they are talking about.
But there are others who have a clue -- even some Climate Scientists -- who can (and do) nonetheless come to the wrong conclusion based on incorrect logic and/or mathematical analysis.
For example, Climate Scientist Roy Spencer recently posted an analysis of the isotope issue “More CO2 peculiarities: The C13/C12 isotope ratio” ), which he concluded with the following question:
“BOTTOM LINE: If the C13/C12 relationship during NATURAL inter-annual variability is the same as that found for the trends, how can people claim that the trend signal is MANMADE??”
The blogging mathematician Tamino found a show-stopping mathematical
error "fallacy" (indicating a failure to understand a very basic concept) in Spencer’s analysis (detailed in Tamino's post
A Bag of Hammers) that renders invalid Spencer's central point (implied by the above question).
Below, Horatio would like to consider a different part of Spencer’s analysis, namely his value for the observed ratio
of what Spencer refers to as "C13 variability to C12 variability". In other words, for the atmosphere, the observed ratio of the "time rate of change of concentration of carbon-13-containing CO2 " to the "time rate of change of concentration of carbon-12-containing CO2 " (referred to by Spencer and below as
dC13/dt / dC12/dt)
After some analysis of his own (that has really taxed Horatio's little mousy brain to its maze-running limits), Horatio got a significantly different result for the ratio dC13/dt / dC12/dt than the one given by Spencer on his graph (from “More CO2 peculiarities: The C13/C12 isotope ratio” )
Spencer's graph above is purported (by Spencer) to be for dC13/dt / dC12/dt, for which Spencer gives the (central) value of 0.010952. He elaborates under his graph that “The slope of this line (1.0952%) represents the ratio of C13 variability to C12 variability associated with the trend signals. “
But there is reason to doubt Spencer’s result.
Though almost certainly not the method used by Spencer, an expression for “ dC13/dt / dC12/dt” can be obtained by differentiating both sides of the equation for “delta13CO2” (referred to as “deltaC13” below). This method yields a DIFFERENT RESULT for the dC13/dt / dC12/dt ratio: namely, 0.011084 +- 0.000022 (mean +- 1 std dev) .
Take the time derivative of both sides of the equation for delta13CO2:
d/dt {deltaC13} = d/dt { 1000[ (C13/C12 )/ S – 1] }
where C13 represents CO2 containing C13, C12 represents CO2 containing C12, S = 0.0112372 (C13/C12 ratio for standard) . Note: what has been called "deltaC13" above is given as "13CO2" for the data found here (World Data center for Greenhouse Gases) under "13CO2(flask)"
If one does this and rearranges terms, one finds
dC13/dt /dC12/dt = { (d_deltaC13/dt)(C12)( S) / (1000*dC12/dt) } + C13/C12
where d_deltaC13/dt = time rate of change of “deltaC13”. The value of “d_deltaC13/dt” in the above differential equation was approximated by the slope of the trend of the Mauna Loa deltaC13 data over the period 1990 – 2005.
"d_deltaC13/dt" comes out -0.0249/yr when one trends the annual averages (average of monthly values) for "deltaC13" (ie, trend of annual averages of monthly "13CO2" values found here (World Data center for Greenhouse Gases) under "13CO2(flask)" ). It’s important to note that “d_deltaC13/dt” is negative, so the magnitude of the first expression on the right above must be subtracted from the C13/C12 ratio to yield the ratio “dC13/dt / dC12/dt” .
Values for the other quantities can be obtained/calculated from the Mauna Loa ( "13CO2(flask)" and "CO2(flask)" and annual mean growth rate) data for each of the years.
Plugging in the values of the relevant quantities (calculated from the Mauna Loa data, found here (under "13CO2(flask)" and "CO2(flask)") and yearly increases found here (under "annual mean growth rate") for individual years over the period 1990- 2005, yields a value for the dC13/dt / dC12/dt ratio for each of those years.
The dC13/dt / dC12/dt ratio obtained with this method turns out to be 0.011084 +- 0.000022 (mean +- 1 std dev) -- IE, significantly different than the mean value claimed by Spencer (0.010952).
Specifically, the mean value obtained with the method described above (0.011084) lies outside the 2-sigma range for Spencer’s ratio, (between 0.010912 and 0.010992, assuming his error bar of 0.000020 is 1-sigma) In fact, it lies over 6 sigma(!) away from the value given by Spencer.
As an illustration, with the above method, we can calculate dC13/dt / dC12/dt for the years at the opposite ends of the interval 1990- 2005.
For 1990:
deltaC13(ave) = -7.82417
d_deltaC13/dt = -0.0249
Total CO2 increase = 1.31ppm , so increase in C12 ~= 0.99(1.31) = 1.297ppm
Total CO2 concentration (ave) = 354.2033ppm so C12 ~= 0.99(354.2033) = 350.66ppm
C13/C12 = 0.011149278
So dC13/dt / dC12/dt = (-0.0249)(350.66)(0.0112372)/ { (1000)(1.297)} + 0.011149278
= -0.0000756 + 0.011149278
= 0.0110737
For 2005:
deltaC13(ave) = -8.20083
d_deltaC13/dt = -0.0249
Total CO2 increase = 2.53ppm , so increase in C12 ~= 0.99(2.53) = 2.505ppm
Total CO2 concentration (ave) = 379.9708ppm, so C12 ~= 0.99(379.9708) = 376.17ppm
C13/C12 = 0.011145046
So dC13/dt / dC12/dt = (-0.0249)(376.17)(0.0112372)/{ (1000)(2.505) } + 0.011145046
= -0.000042018 + + 0.011145046
= 0.0111030
In words, the "dC13/dt / dC12/dt" ratios for 1990 and 2005 ( 0.011074 and 0.011103, resp.) obtained with the above method are completely inconsistent with the range specified by Spencer 0.010952 +- 0.000020.
Important Note: the above assumed that one obtains the C12 values from the total CO2 values (for concentration and concentration change) by multiplying the total CO2 value by 0.99. For CO2 concentration, this assumption is indeed very good, since the C12 fraction of the total atmospheric CO2 IS 0.99 (the 13Co2delta of roughly -8 permil is equivalent to a C12 fraction of 0.989) .
And for the yearly change in CO2, this is in all likelihood ALSO a very good assumption, since for the important sources of CO2 (fossil fuel burning, decay and burning of plants, volcanic eruptions, deep ocean, surface ocean, even the standard), C12 still makes up 99% of the total carbon. Furthermore, a relatively small change to the dC12/dt value in the above formula yields virtually the same result. A small change would NOT bring the number into line with the number Spencer obtained, at any rate.
In order for the above analysis to yield the value Spencer got (0.010952), the values involving C12 above would have to differ by a rather large amount from the actual observed yearly changes for total CO2. For example, for 2005, in order to give the answer Spencer got with the above formula, one would actually have to assume that 12CO2 had changed in 2005 by only 0.54ppm as compared to the actual total measured CO2 change: 2.53ppm. That’s simply not plausible.
Note: As a check on the result obtained by differentiating the equation for delta, one can also estimate the ratio dC13/dt / dC12/dt by calculating the total change in C13 (ie, total change in concentration of CO2 containing C13) divided by total change in C12 (ie change in 12CO2) over the entire interval 1990-2005. For the long interval (1990 – 2005), trend lines can be fit to the (Mauna loa CO2 and deltaC13) data and used to determine change in C13 and change in C12. The value that this yields for dC13/dt / dC12/dt is virtually identical with the result obtained with the differential method (which is actually to be expected given the (almost) linear nature of the CO2 and delta13CO2 curves over the period in question – ie, of the annual average data, sans the seasonal variation)
The following graphs (added Feb 7) illustrate these (nearly) linear relationships.
So, the real question is this: what is the source of the significant discrepancy between Spencer’s result (0.010952 +- 0.000020) and the value obtained above with the differential method above ( 0.011084 +- 0.000022)?
Assume for the moment that the above analysis is correct. This makes Horatio wonder if Spencer might possibly have calculated a different ratio than the one he listed on his graph: namely, to wonder whether he is actually displaying dC13/dt / dCtotal/dt (where dCtotal = dC13 + dC12) instead of dC13/dt / dC12/dt as he claims.
Interestingly, if that were indeed what Spencer has done, then the actual (ie corresponding) value for dC13/dt / dC12/dt would be 0.011075, which would be consistent with the value obtained with the method above (0.011084 +- 0.000022). In fact, 0.011075 is only about 0.5 sigma away from 0.011084. But this is just speculation, of course. One would have to see Spencer’s actual calculations to know.
A final note: if the 0.011084 value is correct, that would STILL imply that the CO2 added to the atmosphere over the period 1990-2005 was significantly depleted in C13. The corresponding delta for 0.011084 is about -13.6 permil, but one must be careful NOT to assume that this is the delta for any specific CO2 added to the atmosphere for any specific source.
There is no reason to expect that the observed dC13/dt / dC12/dt ratio for the atmosphere over time would be as close to the mean C13/C12 ratio for fossil fuel (ie, 13CO2/12CO2 ratio for CO2 produced by burning of fossil fuel, mean ~= 0.010956) as Spencer's number (0.010952) is, at any rate.
Even if one assumes that fossil fuel (mean delta ~= -25) burning has been completely responsible for the increase in atmospheric CO2 over the period 1990-2005 (and, notwithstanding claims to the contrary, there is little reason to suspect that it was not), there is still a certain amount of CO2 interchange* -- eg, with the ocean -- going on and the C13/C12 ratio for CO2 dissolved in the ocean is greater than that for fossil fuel. It seems that even a relatively small amount of interchange would tend to boost the observed dC13/dt /dC12/dt ratio over that which would be observed in the total absence of any such interchange.
*Update May 6, 2009: Horatio looked into the isotope issue in a little more detail in this post and the "interchange" referred to above indeed accounts for the difference between the observed C13/C12 ratio of the "accumulated" CO2 over the period 1990-2005 (delta ~= -13.6 permil) and the " use weighted" mean C13/C12 ratio of the CO2 produced by fossil fuel combustion (delta ~= -28 permil)
Update (added Feb 27, 2009)
Not surprisingly, scientists who research (and teach about) carbon isotope analysis have calculated the "delta13C" (referred to above as "deltaC13" and "delta13CO2") of the CO2 that has accumulated in the atmosphere over time ("delta13Cacc").
The value given by the course author(s) for "delta13Cacc" is -13.6 permil, which they are quick to note, is not equal to -28 permil, (the "use weighted" mean delta for CO2 produced from fossil fuel burning). See course notes below or click on above link (number "5", letter "C": "The Global 13CO2 Cycle -- The Anthropogenic Perturbation")
Horatio made the same basic observation above, though he gave a mean delta for CO2 from fossil fuel burning that was slightly different (-25 permil).
As noted in the post above, if one calculates the corresponding delta for a C13/C12 ratio of 0.011084 (the mean value Horatio calculated above for "dC13/dt / dC12/dt" over the 1990-2005 period) one gets, lo and behold, a delta for the accumulated CO2 of -13.6 permil.
In other words, the result is the same in both cases.
Mere coincidence??
Perhaps**
(**A "mass and isotope balance" between the years 1990 and 2005 yields -13.4 permil, essentially the same result. For details, see Update 2, below the lecture notes)
The time period considered is significantly different (with all that entails), but then again, in both cases, fossil fuel combustion-generated CO2 (with virtually the same delta13C value) has been added to the atmosphere in significant quantity -- and, notwithstanding claims to the contrary, been responsible for the vast majority of the overall CO2 increase. Also, in both cases, the very same processes have governed the interaction of atmospheric CO2 with the other parts of the climate system over time (eg, interchange of CO2 with oceans and with green plants).
Horatio is a bit surprised by the fact that the results for the different periods are identical out to one place past the decimal point (and Horatio takes this with a grain of salt).
But Horatio is not particularly surprised that the two delta13Cacc values would be similar, nor by the fact that they would both differ significantly from the mean delta13C for CO2 produced by burning fossil fuel.
On the other hand, Horatio wonders whether climate scientist Roy Spencer might be just a little bit surprised by all this. After all, Spencer claimed a mean value of 0.010952 for dC13/dt / dC12/dt" (1990-2005) and pointed out the proximity of his value to the (C13/C12) value for CO2 from fossil fuel combustion. A C13/C12 ratio of 0.010952 (Spencer's number) yields a delta13C of -25.4 permil which is close to the use weighted mean delta for fossil fuel combustion (-28 permil*) .
But... -25.4 permil is significantly different from the delta13C for accumulated CO2, given by the authors of those course notes (and by Horatio): -13.6 permil. And those authors certainly seem to make a point that is the direct opposite of the one made by Spencer, with their question "Why doesn't the delta13C of the accumulated CO2 equal -28 permil if the increase in CO2 concentration is due to fossil fuel combustion?"
*The information Spencer provides (in his blog post) yields/implies a delta13C for fossil fuel combustion of - 26.0 permil ( ie, for C13/C12 ratio of 0.010945), "The slope of this line (1.0952%) represents the ratio of C13 variability to C12 variability associated with the trend signals. When we compare this to what is to be expected from pure fossil CO2 (1.0945%), it is very close indeed: 97.5% of the way from “natural” C13 content (1.12372%) to the fossil content." -- Roy Spencer
**Update2 (added March 12, 2009)
Using the annual mean (mean of monthly values) of delta13C and (total) CO2 concentration for 1990 and 2005 (obtained from the Mauna Loa data found here (under "13CO2(flask)" and "CO2(flask)" and given in the "illustration" above), a "mass and isotope balance" (see "lecture notes") yields a value for delta13Cacc (delta13C for accumulated CO2) for the period 1990-2005:
(-7.82417)(354.20ppm) + (delta13Cacc) (25.77ppm) = (-8.20083)(379.97ppm)
which gives
delta13Cacc ~= -13.4 permil
This is essentially the same as the value obtained with the differential method above: -13.6 permil (which is the same as the value given in the lecture notes, albeit for a different period of time).
The slight difference between -13.4 and -13.6 is due not only to the different methods used, but also to the fact that the "isotope and mass" budget was performed using values for delta13C and (total) CO2 for only two years (1990 and 2005, at the ends of the time interval) while the -13.6 permil value obtained by Horatio above was an average for all the years in the 1990-2005 interval. The nearly linear nature of the delta13C and CO2 graphs (see above) makes the two results close (but not precisely) the same.
