*by* Carl Edward Rasmussen, 2018-12-06

An excellent source of recent historical monthly measurements of the atmospheric concentration of carbon dioxide at Mauna Loa in Hawaii is available at this site, the actual data being available here. Thank you to the National Oceanic and Atmospheric Administration (NOAA) for providing this. The data set contains monthly average measurements from 1958 until today, figure 1:

The monthly average carbon dioxide concentration in parts per million [ppm] is plotted as a function of calendar year in red dots, the inset shows a closeup of the most recent years. This graph is sometimes known as the Keeling Curve. The data shows an increasing concentration of carbon dioxide from about 315 ppm in 1958 to about 408 ppm in 2018. For context, the atmospheric carbon dioxide concentration remained constant at about 280 ppm for the 10000 years leading up to the industrial revolution. The Mauna Loa carbon dioxide concentration is not identical to the entire atmospheric average, but it's pretty close (there is a small dependency on latitude).

There is a fairly pronounced seasonal fluctuation, which is caused by plants taking up more carbon dioxide in the summer than in winter. Here, we're not primarily interested in the seasonal variation, but we will still have to understand what it looks like, in order to remove its effects from the data. The seasonal component itself looks like this, figure 2:

The seasonal variation as a function of calendar month and year, the coloured equi-concentration curves are labeled with the magnitude of the contribution in ppm. There is a wrap-around boundary condition between December and January. The seasonal component shows a fairly rapid decrease in the four and a half monts between middle of May and end of September or early October (when photosynthesis is most active in the northern hemisphere), followed by a slower rise over seven and a half months to the middle of May. The seasonal component is itself slowly changing, becoming more pronounced with time, and moving earlier in the year. The peak to peak magnitude has grown from 5.8 ppm in 1958 to 7 ppm in 2017. The average of the seasonal component over a year is always zero.

In figure 1, a de-seasonalised blue region has been superimposed on the data. The (small) width of the blue area indicates the 95% confidence for the underlying de-seasonalised value. Since we are only basing our inferences about carbon dioxide concentration on finitely many, slightly noisy measurements, we can't be absolutely sure exactly what the concentration is at a certain time. But we can be 95% confident that the actual value is within the blue band.

What is the growth of carbon dioxide in the graph? The growth rate is the instantaneous slope of the de-seasonalised curve (also called the derivative) and is measured in parts per million per year [ppm/y]. However, the raw instantaneous growth rate isn't that useful to plot for two reasons: firstly, the instantaneous growth rate is not itself that interesting, because it doesn't reveal much about how the carbon dioxide concentration is evolving over finite intervals of time, and secondly, the instantaneous growth rate is both very variable and associated with a large amount of statistical uncertainty, as our data only contains monthly measurements. To overcome this issue, we either 1) first compute the instantaneous growth rate and then average the result locally in time, or 2) first average the concentration values locally in time, and then compute the growth rates. In fact these two views are mathematically equivalent, so you can use whichever interpretation you prefer. The resulting growth rates are shown in figure 3:

The growth rate of the de-seasonalised carbon dioxide concentration for three different time averages, annual in blue, three-yearly in green and decadal in red. The annual averaged growth rate tends to wobble about a fair bit. This is probably caused by random fluctuations in wind patterns leading to local effects of mixing of air volumes across altitude and latitude and to other short term variations in carbon dioxide fluxes. These effects are quite short term, typically lasting just a year or two. The annual averaged growth-rate is also associated with a fair bit of uncertainty, indicated by the width of the blue region, which corresponds to the 95% confidence region. Note, that the uncertainty about decadal averaged growth rate is much smaller than the annual averaged growth rate (because it's based on more measurements). Note also, that the band is wider towards the boundaries of the data, reflecting that we're less certain about the rate here. In 1960-1970 the growth rate was slightly less than about 1 ppm/y, but the growth-rate has been steadily increasing, reaching 2.43±0.27 ppm/y (mean ± 2 std dev) towards the end of 2018. This means that currently, the concentration of carbon dioxide is growing by about 2.4 ppm per year.

The National Oceanographic and Atmospheric Administration (NOAA) also provide a graph of carbon dioxide growth, which looks similar to the plots presented here. The difference is that the plots presented here emphasises the continuous nature of the process (rather than discrete yearly, or decade-long average growth events), and they quantify uncertainty.

So, what does the data tell us? It shows that all is not well in the state of the atmosphere! In order to prevent further warming, the carbon dioxide levels must not grow any further. On the growth curve, this corresponds the curve having to settle down to 0 ppm/y. There is absolutely no hint in the data that this is happening. On the contrary, the rate of growth is itself growing, having now reached about 2.3 ppm/y the highest growth rate ever seen in modern times. This is not just a "business as usual" scenario, it is worse than that, we're actually moving backward, becoming more and more unsustainable with every year. This shows unequivocally that the efforts undertaken so-far to limit green house gases such as carbon dioxide are woefully inadequate.

Unfortunately, the carbon dioxide data has been subject to some
misleading interpretations, due to poor statistical reasoning. For
example in the Nature Communications paper
[Keenan et al,
2016] entitled "Recent pause in the growth rate of atmospheric
CO_{2} due to enhanced terrestrial carbon uptake", the authors
identify "a pause in the growth rate of atmospheric CO_{2}",
lasting from 2002 to (at least) 2014. They also identify a "point of
structural change" in the growth rate in 2002. In fact it is highly
unlikely that any such pause or point of structural change actually
exists. On the growth rate figure above, the decadal average growth
rate was 1.93±0.01 ppm/y at the start of 2002 and
2.40±0.06 ppm/y at the end of 2014 (mean ± 2 std dev),
so the growth rate has indeed grown. In fact, the growth rate of the
growth rate (also called the acceleration) within the start of 2002 to
end of 2014 interval is 0.0363±0.0048 ppm/y^{2}, whose
lower bound is beyond the upper bound for the average acceleration in the
40 years prior to that (1962 to 2002) at 0.0295±0.0016
ppm/y^{2}, showing that it is very likely that the growth rate
has in fact grown faster in the 2002 to 2014 interval, than during the
40 years prior to that. Publishing such flippant ideas should be
avoided, as they create confusion where there should be clarity. It
would be prudent of the authors to retract their paper (or explain why
it is a valuable contribution when it doesn't agree with
observations).

It is indisputable, that in the long term, the carbon dioxide growth must be brought down to zero, otherwise the earth will just keep getting warmer. However, it is unrealistic to assume that carbon dioxide growth can be halted instantaneously. It is more realistic to assume that the reduction in the growth will take place gradually. The level at which the concentration will stop growing depends on the schedule of the growth reduction. Let's assume that the global growth will be reduced at a fixed relative rate, at some percentage per year, indefinitely. Below is a figure showing what the equilibrium carbon dioxide concentration will be depending on the rate of growth reduction:

The graph shows that the faster the growth rate is reduced, the
lower the final concentration. To limit the final level to be twice
the preindustrial level will require roughly an annual 1.5% reduction
in the growth rate if we start now; if we first wait 25 years with
taking action, close to 3.5% reduction per year will be required. A
doubling of the carbon dioxide concentration is estimated to cause a
warming of between 1.5°C-4.5°C, see IPCC
5th assessment report, and see also the concept of climate
sensitivity. Reduction rates of less than about 0.3% will lead to
very large concentrations. (But note that currently the growth rate is
*growing* by about 1.3% per year, not reducing at all.)