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Mauna Loa carbon dioxide: a fit

I wanted to find a nice function that gives a satisfactory description of the carbon dioxide concentration as a function of time.

Mauna Loa, one of the five volcanoes underlying Hawaii, a middle Pacific state of the U.S., is the most standardized place where the concentration is measured. See this NOAA page on Mauna Loa weekly and the raw data I have used.

If you open the page with the raw data, you will get over 2,000 weekly concentrations of carbon dioxide in ppmv (parts per million of volume – or equivalently, parts per million of the number of molecules) between May 1974 and February 2013. The page usefully offers you the decimal year. For example, February 10th, 2013 is written as \[

2013.1110 = 2013+\frac{40.5}{365}

\] because February 10th is the 41st day of the year with 365 days and the 41st day is offset by the average between 40 (number of days before it) and 41 (the ID). Think about it, it's natural: the extra one-half labels the noon, the middle of the day. The next step is to ignore the historical data and the entries –999.99 for the concentrations.

I did various things but the most important result I want to show you is a nonlinear fit for the concentration as a function of \(y\), the year that includes the fractional part remembering the date. Note that in the past, I offered you my "exponentially increasing" formula for the carbon dioxide concentration as a function of the year, ignoring the date – the seasonal variation:\[

c = 280 + 22.3 \exp\left(\frac{{\rm year}-1920}{57}\right)

\] However, we want to include the seasonally oscillating part of the graph. The following nonlinear fit is self-explanatory; all the "complicated" real numbers in it (280 isn't one of them) were optimized by the NonlinearModelFit command in Mathematica.\[

{\rm conc}(x) &= 280 + \exp(4.359 + 0.02088 x)\\
&- 0.8263 \cos[2\pi x] + \\
&+ 0.6033 \cos[4\pi x] + 0.0255 \cos[6\pi x] - \\
&- 0.0208 \cos[8\pi x] + 0.0166 \cos[10\pi x] - \\
&- 0.0135 \cos[12\pi x] + \\
&+ 2.8359 \sin[2\pi x] - \\
&- 0.5413 \sin[4\pi x] - 0.0957 \sin[6\pi x] + \\
&+ 0.0430 \sin[8\pi x] + 0.0218 \sin[10\pi x] - \\
&- 0.0002 \sin[12\pi x],\\
x &= y - 1994.

\] Sorry, for the strange brackets and other imperfections of the \(\rm \LaTeX\). You see that the base concentration was chosen to be 280 ppm and the increase was prescribed to be exponential. In the exponent, the slope is\[

0.0208774 = 1/47.899

\] so the \(e\)-folding time seems to be shorter than those 57 years that are optimal for the interpolation from the beginning of the Industrial Revolution. You could say that according to the 1974-2013 data, it seems that the time in which the CO2 excess above 280 ppm gets multiplied by \(e=2.718\dots\) is shorter than 50 years. Roughly speaking, the same thing holds for the annual emissions. Here is the 1974-2014 graph.

But what I find interesting is the dependence on the date or seasons, i.e. the fractional part of the year \(y\). My nonlinear fit was a general Fourier expansion up to the 6th higher harmonics – 12 coefficients in total (six for cosines, six for sines). For your convenience, here are two cycles of the cosine/sine part of the fit:

Apologies that the last, negative half-wave isn't filled black. I don't want to spend hours with similar things. The shape of the curve hasn't substantially changed when I added the 6th harmonic cosines and sines. But you see that it dramatically deviates from a simple shifted sine. It's a rather complicated curve. Incidentally, I have checked that I was visually unable to see the difference between the seasonal curve (fit) extracted from the first 20 years of Mauna Loa and the last 20 years of the dataset.

A naive person could think that the Northern and Southern hemispheres should cancel and there shouldn't be a difference between springs and autumns. However, that's wrong because what's primarily responsible for the seasonal variations are life processes on land and the land masses are mostly on the Northern Hemisphere (Eurasia and North America are large). So the graph pretty much emulates what the Northern Hemispheres wants to do to the CO2 while our friends in Australia and Antarctica are just negligible parrots, penguins, and kangaroos who are emulating what we're doing half a year later. ;-)

During winter, the plants – the players that are capable of absorbing CO2 – are losing their vitality and activity so the CO2 is increasing up to the maximum near 3.12 ppm (plus the long-term running average) for \(x\in 0.36+\ZZ\) – something like May 10th. That's where the trend is reverted and Northern Hemisphere plants start to blossom and absorb more CO2 than the annual average. That's why the CO2 drops up to the minimum around –3.58 ppm (plus the long-term running average) for \(x\in 0.74+\ZZ\), something like September 27th. Then the seasonal CO2 anomaly starts to go up again, and so on.

What I also find interesting is the slight "hole" near \(x\in 0.15+\ZZ\). The graph really refuses to be smooth at that point. Also, you may notice that \(0.74-0.36=0.38\) is substantially less than \(0.5\) which means that the decrease of CO2 during the Northern summers is faster than the increase of CO2 in the rest of the year. In other words, when plants start to boom and consume CO2, they may do so more quickly and efficiently than when they're failing to boom. ;-)

The latest reading is 396.74 from February 10th, 2013. In 2013, this is bound to increase by 3.6 ppm or so by May 10th or so. In other words, the maximum of Mauna Loa CO2 for 2013 in May will slightly exceed 400 ppm for the first time around mid May – about 400.3 ppm will be the annual maximum. I guess that we will hear about it again. If the alarmists weren't preparing this story for May 2013 yet, one of them who is a TRF reader at the same moment will surely steal the idea from your humble correspondent!

The sine-and-cosine part of the fit (see the last graph with two cycles) vanishes around January 8th and July 20th which are the dates for which the actual measured concentrations may be interpreted as the "long-term running means" with the seasonal variations removed. On July 22nd, 2012, they had 393.98 pm. On January 6th, 2013, it was 395.53 ppm. The latter is close to what we have "now" when the seasonal variations are suppressed.

Let me mention that between May and September, the seasonal variations contribute the drop by \(3.12+3.58=6.70\) ppm of CO2. If you could make plants thrive in the winter as well, you could easily subtract something like 13 ppm of CO2 from the atmosphere every year, well above the 2 ppm by which we are increasing the concentration every year (it's 1/2 of 4 ppm we are adding; the other half is already being absorbed by the enhanced consumption of CO2 due to the elevated concentrations).

Did you know that just 6,500 years ago, Britain was connected to Europe in this way? The British Euroskeptics were weaker than today. Hunters were working hard in the land just in between Britain and Denmark. On Wednesday night, I didn't want to believe those claims – I was imagining that the author of the story was envisioning some huge continental drift in just thousands of years ;-) – but when you think about it, it's completely natural that there had to be a land, Doggerland, because big chunks of the Northern Sea are extremely shallow, around 30 meters, and the sea level jumped by more than 100 meters in the last 20,000 years as the continental ice sheets melted.

During the next ice age, sometimes around 60,000 AD, Doggerland could become a land again. It would be fun to know what kind of people (or creatures) will be living there and in what way they will be communicating. Alternatively, we may rebuild Doggerland by landfills and other constructions much earlier, perhaps in 2020 AD.

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snail feedback (17) :

reader AJ said...

I was using similar equations on the anomalies to estimate when the rate would go from accelerating to constant. I figured sometime in the second half of this century. Then I looked at the rates over the last few decades. IIRC, we've seen a deceleration over the last decade. I speculated that this was either due to the financial crises or the boom in natural gas, so this deceleration could be either short or long term. Just goes to show the limits of curve fitting.

reader Martin Clark said...

I know that Mauna Loa measurements are used because they are longest. I have from time to time measured CO2 levels for similar reasons - evidence of something happening. Not volcanic activity, but inadequate air movement, particularly in confined workspaces. Not because of the gas itself, but because of what its elevated level indicates.

What puzzles me is why there is apparently so little monitoring going on, given that 1% dataloggers are cheap and accurate these days? Around here at least, every cold room and every bar has to have at least one.

I have turned my interest to the outside CO2, and started to learn a bit more. I am located close to a west Pacific shoreline. Occasionally, I have detected CO2 "spikes" during late summer afternoons with (unmitigated) airflow coming off the ocean. 'Carbon pipe' outgassing? Can't think of anything else ...

Sure, the level goes up after sunset, from about 395-400 to 425-450, but it does that EVERY day here - outgassing from vegetation.

Haven't done any regular logging, but my rig has detected:

- Bushfires (4km) and controlled burning (13km) upwind.
- A diesel truck idling out in the road (10m)
( but not one passing 20m away)

- Me passing next to the sensor with a lawn-mower (not cussing, hold my breath, so as not to skew the results).

reader Edim said...

Annual CO2 cycle is caused by the annual SST cycle.

The accumulation of atmospheric CO2 is also caused by the annual cycle, which is pumping CO2 into the atmosphere. The annual flow of this pump is dependent on the temperature level - lower the temperature, lower the flow, and at sufficiently low temperatures the flow will reverse.

reader Gene Day said...

Nice post, dougproctor. If everyone engaged in such careful thinking we would not have climate alarmism, would we?.
Re. your cited Climate-of-the-Past article, it seems fanciful to even think that any local temperature record might reflect the global average, never mind the author’s obvious rationalizations in support of alarmism. Junk science is, alas, junk science.

reader Gene Day said...

If civilization had developed 6500 years earlier one can imagine the climate alarmism. With the seas rising at about 4 meters per century (16 inches per decade) the AGW crowd would have had a field day!

I would add a note: The subsidence and uplift of land masses are very steady processes; i.e. the accelerations are generally negligible. Those alarmists worrying about rising seas are actually talking about upward acceleration, sea level advances above the slow, continuous (but decreasing) rise that has been going on throughout the current interglacial period. Tidal gauge measurements of the rise rate (which averages around 1.7 mm/year, globally) are a bit clouded by these steady tectonic effects but the resulting data are very sensitive to changes in acceleration. It seems apparent to me that any anthropological contribution to our rising seas is less than about 0.1 mm/year = 0.4 inches per century despite a 45% increase in atmospheric CO2.

reader dougproctor said...

Thanks for the supportive comments. Critical thinking is, alas, neither common nor easy. Each time I think I have thought something through, indeed, I am soon brought up by either another thought or another's thought of something I missed. It might be pertinent, or it might not be pertinent, but I is generally worth considering.

Critical thinking requires background knowledge, sometimes arcane, an imagination, and a willing skepticism - to what you think yourself. The liberal warmist seems to be lacking in all three items, preferring the authority of respected others and the belief that details don't matter. Critical thinking also requires the acceptance that one can be wrong, dead wrong, even with the best of intentions. The liberal warmist would see this "error" incompatible with the best of intentions, and so a very difficult thing to accept.

reader dougproctor said...

So you are saying the actual cycle is SST related, not biotic? If so, what about the longer-term release? Should the Mauna Loa CO2 record be adjusted downward when we want to identify anthropogenic influences?

I thought there might be an SST contribution, but I never thought it might be dominant, just a non-corrected contributor.

reader dougproctor said...

At one point I tried to figure out the relationship between A-CO2 emissions and CO2 increases in atmospheric content. The amount required seemed to increase with time, which I wondered was a result of WWF et al influences in over-estimating emissions for political and alarmist reasons (back to the benefit-of-the-doubt concept I discussed above). If you could see a reduction in CO2 rising wrt emissions, I figured it was an argument for enhanced uptake biologically and chemically - either the ocean or rock (carbonates). Or we could be just seeing the results of the actual residence time of CO2 in the atmosphere.

when you first start to produce an oil field, like the North Dakota Bakken, production rate increases rapidly. That would be the start of CO2 into the atmosphere causing a rapid uptick in pCO2. Then, with production behind you, your future production gains are offset by production declines in the initial wells. That would be the removal rate, i.e. residence issue, of CO2. As time goes by, your rate of production increase equals your decline rate, and total production flattens. That would be the new, stabilized CO2 level. Only if you continue to increase production at a rate faster than decline do you get a net production increase; stability, and then decline, is the end result.

For CO2 today, China (more than India) is the dominant player. At 396 ppmv, the removal, or "decline rate" is greater than at 280 and is not stabilized. Unless China continues at increasing its rate of CO2 emissions, it is unlikely that the world will see CO2 rising as before.

I never considered this before: if the residence time is greater than the record span of Mauna Loa, we cannot be at a stabilized CO2 level. We have been contributing CO2 greater than the decline rate, just as the growth of the Bakken production so far has exceeded the decline rate of the gross, preceeding producing wells.

So we need a comparison of decline rate to production rate for CO2 to determine a stabilized future pCO2. Because what we have now is not stabilized!

reader Gene Day said...

I’m not sure that I would use the word “liberal” quite so liberally but I love your eagerness to engage in self-criticism. That is precisely what we need in these trying times. I hope that we will hear more from you in TRF.

reader Alexander Ač said...

Next ice age? Sure, we are sooo powerless to change radiative balance of atmosphere, just as we are unable to pollute oceans, and drive to extinction most of the big animals... yes, we are nothing compared to Mother Nature. Or not.

reader Luboš Motl said...

Dear Alexander, we are surely powerless with respect to ice ages when we restrict ourselves to commonly used technologies. Ice ages represent something like 8 °C drop of the global mean temperature relatively to interglacials.

Not even the most insane alarmists claim that the human activity may change the temperatures globally by a similar or greater amount - with possible alarmists who are already confined into mental asylums.

In 60,000 years from now, when the simulations predict a new cold peak of the next ice age, if the people are still around, chances are that they will be capable of making things like changing the global mean temperature by 8 °C by a safe enough technology. That doesn't mean that they will do it or that it's the right thing to do it. I suspect that to achieve certain things, they will still change the "climate" only locally - effectively, they will keep many things inside state-of-the-art "greenhouses".

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reader AJ said...

Agreed that neither the yearly emissions nor the air concentration have stabilized. Both are still increasing. My point was that the slope of the yearly increase in concentration over the last ten years was insignificantly negative, I'll guess mostly due to the financial crisis, but I wouldn't exclude a long term trend either.

If the rate of increase has stabilized then that would be great news. Given the logarithmic relation, the rate of radiative forcing would be decelerating.

reader dougproctor said...

What I was thinking was that if you add CO2 faster than it can be removed, the CO2 will rise to a point higher than it will stay at at that input level once the removal process kicks in. So if CO2 emissions stabilized, the ppmv pCO2 we currently have will drop.

Dump salt into a beaker of hot water fast enough, solid salt builds up even though later all the salt dissolves. For CO2, the reverse.

I don't know if this is true, has to do with the input vs removal rates, but the important thing is that we are probably NOT at a final ppmv pCO2 based on the amount of CO2 emissions to-date.

And, again, on a separate note, we need to adjust the current ppmv for SST changes as, in a reverse sense, the IPCC does for sea-levels wrt GIA.

reader Broadlands said...

Updating the 1987 data of Newell and Marcus "Carbon Dioxide and People", there remains a very strong quadratic curvilinear correlation between global population and Mauna Loa CO2 from 1959 through 2012. r-squared: 0.9992. Using current world population figures a projection to 400 ppm reveals that it doesn't take place until some time in 2015.

reader Bart said...

Atmospheric CO2 is almost completely determined by global temperatures. The relationship is:

dCO2/dt = k*(T - To)

dCO2/dt = CO2 rate of change
k = coupling factor in ppm/degC/time-unit
T = global average temperature
To = equilibrium temperature at current state

This type of relationship can be expected when you have continuous flows in and out of the surface system which are modulated by temperature. The affine relationship above is likely a linearization of a more complicated function relationship which varies over time.

With this equation, you can reconstruct the CO2 level at any time since reliable measurements became available using temperatures only - human inputs are essentially superfluous. Clearly, they are rapidly sequestered by natural processes.

You can use any of the major temperature sets, though the affine parameters vary. The best agreement seems to be with southern hemisphere data, which suggests it is mostly an ocean phenomenon. This is reasonable since ocean circulation serves as a massive conveyor belt continually bringing CO2 into the surface system, and pulling it back out again.

reader Knut Holt said...

Istead of rebuilding Doggerland, I think one wil custruct gigantic floating islands, that also have the capability to move like ships, so that thay allways are at a favourable place on the oceans.