This is a harder, but more comprehensive treatment of the use of 13-carbon to 12-carbon ratios in determining the sources of carbon in the atmosphere. It covers the main points of the evidence that reaches the conclusion that the carbon dioxide entering the atmosphere since the  Industrial Revolution comes mainly from the burning of fossil fuels. 


Factors influencing the trends in the values of the d13C/12C ratio in atmospheric CO2


The following diagram shows the ranges of values of the d13C/12C ratio in important contributors to the carbon cycle.




The recent ‘mix’ of fossil fuels has a d13C/12C value of -28 ‰. It is the burning of fossil fuel that mainly affects the annual decreases in the value of d13C/12C in the atmosphere. A graph of the monthly variation in d13C/12C since 1980 from the Mauna Loa station is shown below.




The trend in the values is -0.00194 ‰ per month or -0.023 ‰ per annum. Superimposed on the general negative trend are the seasonal variations that are produced mainly by the photosynthesis/vegetation decay cycle that operates mainly in the northern hemisphere. There is also considerable interchange of CO2 between the atmosphere and the oceans, but there is only a small discrimination of carbon isotopes involved. Although the isotopic signals of the atmosphere and vegetation are considerably different, there is very little overall change in the d13C/12C value as the carbon taken up as the vegetation grows is replaced when the vegetation decays. The small changes that occur are because there is a lag, sometimes up to 20 years, before carbon from rotting vegetation re-enters the atmosphere, some of which then has a slightly different d13C/12C value.

            Monthly details of the seasonal changes in the CO2 concentration and the corresponding d13C/12C values are shown in the next diagrams for the years 1981 and 2000.





These diagrams show the general inverse relationship between rising CO2 concentration and increasingly more negative d13C/12C values, consistent with the increases in CO2 concentration being associated with the dilution of the 13C content as vegetation decays. Likewise, in the growing season the concentration of CO2 decreases and is accompanied by an inverse rise in the value of d13C/12C as photosynthesis discriminates against the heavier isotope. The maxima and minima of some of the plots are not always synchronized and this might be due to small variations in the participation of the oceans in the seasonal exchange of CO2. In general plots of this kind do indicate that higher concentrations are to be expected to be associated with more negative values of d13C/12C and that is what is observed for the annual changes. Since the seasonal and annual changes are expected to be different because of the different sources of carbon, it is desirable to separate the two effects. These latter are the seasonal variations caused by the growth/decay cycle of the biosphere and the annual variations which are caused by the continual injections of fossil fuel carbon.


Estimation of d13C values


The basis of the Keeling and Miller/Tans plot methods is conservation of mass. The atmospheric concentration of a gas is a combination of the background atmospheric concentration and variable amounts of that gas added by sources such as those from biota, the oceans and from fossil fuels:


CM = CB + CA                                                        (1)


where CM, CB and CA are, respectively, the measured atmospheric CO2 concentration the background CO2 concentration, and the additional concentration component produced by the source, which has raised atmospheric CO2 concentration above background.

Given conservation of mass,


(d13CM)CM = (d13CB)CB + (d13CA)CA                                     (2)


where (d13C) represents the carbon isotope ratio of each CO2 component. Combining equations (1) and (2),


d13CM = CB(d13CB - d13CA)(1/CM) + d13CA                                 (3)              


A plot of d13CM versus 1/CM [Keeling plot] gives a straight line with an intercept equal to d13CA.


Alternatively, the combined equations may be organized to give:


(d13CM)CM = CB(d13CB - d13CA) + (d13CA)CM                             (4)


A plot of (d13CM)CM versus CM [Miller/Tans plot] gives a straight line with a slope equal to d13CA.


An example of a Miller/Tans plot is shown below. The data are for one year with its seasonal changes in CO2 concentration.



The slope of the trend line is -26.6 ‰ [the value of d13CA] and is consistent with the suggestion that the changes are brought about by the seasonal injection and removal of biotic carbon.


The next plot is for the annual means.




The slope [-13.6 ‰] is significantly different from that for the seasonal changes, but needs some further explanation. The annual changes in d13C/12C are affected by the injection of fossil fuel depleted in 13C. They, unlike the seasonal changes which are self-regulatory, alter the atmosphere/ocean equilibrium with respect to the two isotopes as well as the overall equilibrium with respect to the partial pressure of CO2. This is indicated by the above graph, but the perturbation of the isotopic equilibrium is offset by the quite large general exchange that occurs between the ocean and the atmosphere of some 90 Gt C per annum. There is a competition between two opposing processes; the essentially kinetic process that force the discrimination of the two isotopes and the second law of thermodynamics that ensures that to some extent the isotopes are scrambled up between the two phases. The thermodynamic processes that apply to the exchanges between the atmosphere and the land and ocean sinks are known as disequilibria. They occur in a spontaneous attempt to correct the perturbations produced by the injections of 13C-depleted fossil fuel carbon. This explains why the observed d13C  value is not as negative as it would be if all the injected carbon remained in the atmosphere and if the restoring thermodynamic scrambling did not occur.

            These ideas are summarized by the following diagram.





In general terms, process 1 represents what happens when an injection to the atmosphere is made of carbon depleted in the 13C isotope. This must be compared with process 6 which represents the experimental observation. The other processes are those that reduce the effectiveness of the initial injection and are connected with the land biota and the oceans. Process 2 is the effect of atmosphere/land interchange in the restoration of the isotopic equilibrium that has been disturbed by the injection of the 13C-depleted CO2 from fossil fuel burning. Process 3 represents the effect of some of the injected carbon being removed to the land sink. Process 4 is the ‘disequilibrium’ effect with respect to the oceans; exchange with the oceans tends to scramble the isotopes as does process 2 with regard to the land biota. Process 5 completes the ‘circle’ and represents the loss of carbon to the oceans.

            These the details of these processes are in the extensive published literature and show beyond much doubt that the annual decreases in the value of d13C/12C are caused by the continual fossil fuel carbon injections.


The relevant literature is reviewed in Chapters 13 and 14 of Stable Isotopes and Biosphere-Atmosphere Interactions, Eds. L. B. Flanagan, J. R. Ehleringer & D. E. Pataki, Elsevier, 2005 and there are 158 references to the primary journals where the details may be found.