ABOUT SPURIOUS CORRELATIONS AND CAUSATION OF THE CO2  INCREASE


  1. Introduction
  2. The theory
    1. Transient response of CO2 to a step change in temperature
    2. Transient response of CO2 to an increasing temperature slope
    3. Transient response of CO2 to a sinusoid change in temperature
    4. Transient response of CO2 to a double sinusoid change in temperature
    5. Transient response of CO2 to a double sinusoid plus a slope
    6. Transient response of CO2 to a double sinusoid, a slope and emissions
  3. The real world
    1. The cause of the variability
    2. The slopes
    3. The response to temperature variability and human emissions
    4. The derivatives
  4. Conclusion
  5. References


1. Introduction

Some may remember the lively discussions of already several years ago on WUWT about the reasons why I am pretty sure that the CO2 increase in the atmosphere over the past 57 years (direct atmospheric measurements) and 165 years (ice cores and proxies) is man-made [1] [2] [3] [4].  That did provoke hundreds of reactions from a lot of people pro and anti.

Since then I have made a comprehensive overview of all the points made in that series of discussions at:

http://www.ferdinand-engelbeen.be/klimaat/co2_origin.html

There still is one unresolved recurring discussion between mainly Bart/Bartemis and me about one - and only one - alternative natural explanation: if the natural carbon cycle is extremely huge and the sinks are extremely fast, it is -theoretically- possible that the natural cycle dwarfs the human input. That is only possible if the natural cycle increased a fourfold in the same time frame as human emissions (for which is not the slightest indication) and it violates about all known observations. Nevertheless, Bart's (and Dr. Salby's) reasoning is based on a remarkable correlation between temperature variability and the CO2 rate of change variability with similar slopes:


Bart's
                        comparison between T and dCO2/dt

Fig.1: Bart's combination of T and dCO2/dt from WoodForTrees.org


Bart (and Dr. Salby) thinks that the match between variability and slopes (thanks to an arbitrary factor and offset) proves beyond doubt that temperature causes both the variability and slope of the CO2 rate of change. The following will show that variability and slope have nothing in common and temperature is not the cause of the slope in the CO2 rate of change.

Note: for those who know the theory behind transient responses, it gets interesting from point 2.5 onward...




2. The theory.

2.1. Transient response of CO2 to a step change in temperature:

To make it clear we need to show what happens with CO2 if one varies temperature in different ways. CO2 fluxes react immediately on a temperature change, but the reaction on CO2 levels needs time, no matter if that is by rotting vegetation or the ocean surfaces. Moreover, increasing CO2 levels in the atmosphere reduce the CO2 pressure difference between ocean surface and the atmosphere, thereby reducing the average in/out flux, until a certain CO2 level in the atmosphere is reached where in and out fluxes again are equal.

In algebraic form:

dCO2/dt = k2*(k*(T-T0) - ΔpCO2)

Where T0 is the temperature at the start of the change and ΔpCO2 the change in CO2 partial pressure in the atmosphere since the start of the temperature change, where pCO2(atm) was in equilibrium with pCO2(aq) at T0. The transient response in rate of change is directly proportional to the CO2 pressure difference between the pCO2 change in water (caused by a change in temperature) and the CO2 pressure in the atmosphere.

When the new equilibrium is reached, dCO2/dt = 0 and:

k*(T-T0) = ΔpCO2

Where k = ~16 ppmv/C which is the value that Henry's law gives (4-17 ppmv/C in the literature) for the equilibrium between seawater and the atmosphere.

In the next plot we assume the response is from vegetation, mainly in the tropics, as that is a short living response as will be clear from measurements in the real world in chapter 3:

Transient change step

Fig. 2: Response of bio-CO2 on a step change of temperature [10].


As one can see, a step response in temperature gives an initial peak in dCO2/dt rate of change which goes back to zero when CO2 is again in equilibrium with temperature. That equilibrium can be static (for an open bottle of Coke) or dynamic (for the oceans). In the latter case one speaks of a "steady state" equilibrium or a "dynamic equilibrium": still huge exchanges are going on, but the net result is that no CO2 changes are measurable in the atmosphere, as the incoming CO2 fluxes equal the outgoing CO2 fluxes.

Taking into account Henry's law for the solubility of
CO2 in seawater, any in/decrease of 1C has the same effect if you take a closed sample of seawater and let it equilibrate with the above air (static) or have the same in/decrease in (area weighted) average global ocean temperature with global air at steady state (dynamic): about 16 ppmv/C.



2.2. Transient response of
CO2 to an increasing temperature slope:

If the temperature has a slope, CO2 will follow the slope with some delay:


Transient response of
                    CO2 to an increasing temperature

Fig. 3: Response of bio-CO2 on a continuous increase of temperature [10].


A continuous increase of temperature will induce a continuous increase of CO2 with an increasing dCO2/dt until both increases parallel each other and dCO2/dt remains constant. This ends when the "fuel" (like vegetation debris) gets exhausted or the temperature slope ends. In fact, this type of reaction is more applicable to the oceans than on vegetation, but this all is more about the form of the reaction than what causes it...

A typical example is the warming from the depth of a glacial period to an interglacial in the past 800,000 years: it takes about 5,000 years to reach the new maximum temperature and CO2 lags the temperature increase with some 800 +/- 600 years.



2.3. Transient response of
CO2 to a sinusoid change in temperature:

Many changes in nature are random up and down, besides step changes and slopes. Let's first see what happens if the temperature changes with a nice sinus change (a sinusoid):


Transient response CO2
                        to a sinusoid temperature change

Fig. 4: Response of bio-CO2 on a continuous sinusoidal change in temperature [10].


It can be mathematically explained that the lag of the CO2 response is maximum pi/2 or 90 after a sinusoidal temperature change [5]. Another mathematical law is that by taking the derivatives, one shifts the sinusoid forms 90 back in time. The remarkable result in that case is that changes in T synchronize with changes in dCO2/dt, that will be clear if we plot T and dCO2/dt together in next item.




2.4 Transient response of
CO2 to a double sinusoid change in temperature:

To make the temperature changes and their result on CO2 changes a little more realistic, we show here the result of a double sinusoid where the sinusoids have different periods. After all natural changes are not that smooth...:


Transient response CO2
                        to a double sinusoid

Fig. 5: Response of bio-CO2 on a continuous double sinusoidal change in temperature [10].


As one can see, the change in CO2 still follows the same form of the double sinusoid in temperature with a lag. Plotting temperature and dCO2/dt together shows a near 100% fit without lag, which implies that T changes directly cause immediate dCO2/dt changes, but that still says nothing about any influence on a trend. In fact still T changes lead CO2 changes and dT/dt changes lead dCO2/dt changes, but that will be clear in next plot...




2.5 Transient response of
CO2 to a double sinusoid plus a slope:

Now we are getting even more realistic, not only we introduced a lot of variability, we also have added a slight linear increase in temperature. The influence of the latter is not on CO2 from the biosphere (that is an increasing sink with temperature over longer term), but from the oceans with its own amplitude:


Transient CO2 response
                        to a double sinusoid + slope in temperature

Fig. 6: Response of Natural CO2 on a continuous double sinusoid plus slope change in temperature [10].


As one can see, again CO2 follows temperature as well for the sinusoids as for the slope. So does dCO2/dt with a lag after dT/dt, but with a zero trend, as the derivative of a linear trend is a flat line with only some offset from zero.

This proves that the trend in T is not the cause of any trend in dCO2/dt, as the latter is a flat line without a trend. No arbitrary factor can match these two lines, except (near) zero for the temperature trend to match the dCO2/dt trend, but then you erase the amplitudes of the variability...

Thus while the variability in temperature matches the variability in CO2 rate of change, there is no influence at all from the slope in temperature on the slope in CO2 rate of change.

A linear increase in temperature doesn't introduce a slope in the CO2 rate of change at any level.




2.6 Transient response of CO2 to a double sinusoid, a slope and emissions:

All previous plots were about the effect of temperature on the CO2 levels in the atmosphere. Volcanoes and human emissions are additions which are independent of temperature and introduce an extra amount of CO2 in the atmosphere above the temperature dictated dynamic equilibrium. That has its own decay rate. If that is slow enough, CO2 builds up above the equilibrium and if the increase is slightly quadratic, as the human emissions are, that introduces a linear slope in the derivatives.

Transient response CO2
                        to a double sinusoid + slope + emissions

Fig. 7: Response of CO2 on a continuous double sinusoidal + slope change in temperature + emissions [10].


Several important points to be noticed:

- The variability of CO2 in the atmosphere still lags the temperature changes, no matter if taken alone or together with the result of the emissions. No distortion of amplitudes or lag times. Only simple addition of independent results of temperature and emissions.

- The slope of the natural CO2 rate of change still is zero.

- The relative amplitude of the variability is a small factor compared to the huge effect of the emissions.

- The slope and variability of temperature and CO2 rate of change is a near perfect match, despite the fact that the slope is entirely from the slightly quadratic increase of the emissions and the effect of temperature on the slope of the CO2 rate of change is zero...

The "match" between the slope in temperature and the CO2 slope in rate of change is entirely spurious.
 


3. The real world.

3.1 The cause of the variability:

Most of the variability in CO2 rate of change is a response of (tropical) vegetation on (ocean) temperatures, mainly the Amazon.  That the main variability is from vegetation is easily distinguished from the ocean influences, as a change in CO2 releases from the oceans gives a small increase in 13C/12C ratio (δ13C) in atmospheric CO2, while a similar change of CO2 release from vegetation gives a huge, opposite change in δ13C. Here for the period 1991-2012 (regular δ13C measurements at Mauna Loa and other stations started later than CO2 measurements):

d13C
                        - CO2 - temp at Mauno Loa

Fig. 8: 12 month averaged derivatives from temperature and CO2/ δ13C measurements at Mauna Loa [9].


Almost all the year by year variability in CO2 rate of change is a response of (tropical) vegetation on the variability of temperature (and rain patterns). That levels off in 1-3 years either by lack of fuel (organic debris) or by an opposite temperature/moisture change [5]. Over periods longer than 3 years, it is proven from the oxygen balance that the overall biosphere is a net, increasing sink of CO2, the earth is greening [6], [7].

Not only is the net effect of the biological CO2 rate of change completely flat as result of a linear increasing temperature, it is even slightly negative in offset...

The oceans show a CO2 increase in ratio to the temperature increase: per Henry's law about 16 ppmv/C. That means that the ~0.6C increase over the past 57 years is good for ~10 ppmv CO2 increase in the atmosphere that is a flat line with an offset of 0.18 ppmv/year or 0.015 ppmv/month in the above graph.

There is a non-linear component in the ocean surface equilibrium with the atmosphere for a temperature increase, but that gives not more than a 3% error on a change of 1C at the end of the flat trend or a maximum "trend" of 0.00045 ppmv/month after 57 years. That is the only "slope" you get from the influence of temperature on CO2 levels. Almost all of the slope in CO2 rate of change is from the emissions...

The response of CO2 to the temperature variability is certainly from vegetation, but as vegetation is a proven small and increasing sink for CO2, that is not the cause of the increase of CO2 in the atmosphere or the slope in the CO2 derivative.
 

3.2 The slopes:

Human emissions show a slightly quadratic increase over the past 115 years. In the early days more guessed than calculated, in recent decades more and more accurate, based on standardized inventories of fossil fuel sales and burning efficiency. Maybe more underestimated than overestimated, because of the human nature to avoid paying taxes, but rather accurate +/- 0.5 GtC/year or +/- 0.25 ppmv/year.

The increase in the atmosphere was measured in ice cores with an accuracy of 0.12 ppmv (1 sigma) and a resolution (smoothing) of less than a decade over the period 1850-1980 (Law Dome DE-08 cores). CO2 measurements in the atmosphere are better than 0.1 ppmv since 1958 and there is a ~20 year overlap (1960 - 1980) between the ice cores and the atmospheric measurements at Mauna Loa. That gives the following graph for the temperature - emissions - increase in the atmosphere:

temperature, emissions, increase in the
                        atmosphere

Fig. 9: Temperature, CO2 emissions and increase in the atmosphere [9].


While the variability in temperature is high, that is hardly visible in the CO2 variability around the trend, as the amplitudes are not more than 4-5 ppmv/C (maximum +/- 1.5 ppmv) around the trend of more than 90 ppmv. To give a better impression, here a plot of the effect of temperature on the CO2 variability in the period 1985-2000, where several relative large temperature changes can be noticed like the 1991/2 Pinatubo eruption and the 1998 super El Nio:


Wood for trees 1990-2002 T + CO2

Fig. 10: Influence of temperature variability on CO2 variability around the CO2 trend [9]


It is easy to recognize the 90 lag after temperature changes, but the influence of temperature on the variability is small, here calculated with 4 ppmv/C. For the trend, the CO2 increase caused by the 0.2C ocean surface temperature increase in that period is around 3 ppmv of the 22 ppmv measured...

The influence of temperature on the CO2 variability is quite small: +/- 1.5 ppmv around the slope.

 

3.3 The response to temperature variability and human emissions:


With the theoretical transient response of CO2 to temperature in mind, we can calculate the response of vegetation and oceans to the increased temperature and its variability:

RSS transient response CO2

HadCRUT impact on CO2

Fig. 11: Transient response of bio and ocean CO2 to temperature for the satellite (RSS) and near-surface (HadCRUT-SH) trends [9][11].
For the near-surface temperatures the Hadley Center Southern Hemisphere temperatures as these show less deviation with the satellite temperatures (less "adjustments"?).

The bio-response to temperature changes is very fast and zeroes out after a few years [6], the response to the temperature amplitude is about 4-5 ppmv/C, based on the response to the 1991 Pinatubo eruption and the 1998 El Nio.
The response of the ocean surface is slower, but stronger in effect. The 16 ppmv
/C is based on the long-term response in ice cores and Henry's law for the solubility of CO2 in ocean waters (4-17 ppmv /C in the literature).
In reality, both oceans and the biosphere are net sinks for
CO2, due to the increased CO2 pressure in the atmosphere and the biosphere also a net sink due to increased temperature on periods of more than 3 years. That is not taken into account here, but is used in the calculation of the net increase of CO2 in the atmosphere with the introduction of human emissions.

If we introduce human emissions , that gives a quite different picture of the relative dimensions involved:


RSS transient response
                        bio + oceans and emissions

had_cO2 and emissions

Fig. 12: Human emissions + calculated and measured CO2 increase + transient response of bio and ocean CO2 to temperature [9][11].

The influence of temperature both in variability and increase rate is minimal, compared to the effect of human emissions, based on the transient response of oceans and biosphere and the calculated decay rate of human emissions.
The long tau (e-fold decay rate) of human emissions is based on the calculated sink rate (human emissions - increase in the atmosphere) and the increased CO2 pressure in the atmosphere above dynamic equilibrium ("steady state"), which is ~290 ppmv for the current weighted average ocean surface temperature. That is thus ~110 ppmv above steady state and that gives ~2.15 ppmv net sink rate per year. For a linear response, the e-fold decay rate can be calculated:

disturbance / response = decay rate
or for 2012:
110 ppmv / 2.15 ppmv/year = 51.2 years or 614 months.

That the sink process is quite linear can be seen in the similar calculation by Peter Dietze with the figures of 27 years ago [12]:
1988: 60 ppmv, 1.13 ppmv/year, 53 years
Or from earliest accurate CO2 measurements:
1959: 25 ppmv, 0.5 ppmv/year, 50 years

Within the accuracy of the CO2 emission inventories and the natural variability, the decay rate of any extra CO2 above the dynamic equilibrium (whatever the cause) behaves like a linear process...

 

3.4 The derivatives:

What does that show in the derivatives? First the transient response of the biosphere and oceans to temperature variability:

RSS natural response to
                        temperature  

Hadley temp-CO2
                    derivatives

Fig. 13: RSS and HadCRUT-SH temperature changes compared to observed CO2 rate of change and transient response of natural CO2 (biosphere+oceans) rate of change [9][11].


It seems that the amplitude of the natural variability is overblown in the RSS plot, but not in the HadCRUT-SH plot. In both the temperature and the transient response of CO2 are equally synchronized with the observed CO2 rate of change with hardly any slope in the transient response. Thus while all the variability is from the transient response, there is hardly any contribution of oceans or biosphere to the slope in CO2 rate of change.
The overdone amplitude of the natural variability may be because the satellite temperature reacts much faster than the surface compilation, which is the average change of many stations, but that is not that important. The form and timing are the important parts.


Now we can add human emissions into the rate of change:

RSS-transient-emissions-derivative

HadCRUT transinet response + emissions

Fig. 14: RSS and HadCRUT-SH temperature compared to CO2 increase and transient response of natural CO2 + emissions rate of change [9][11].

For an exact match of the slopes of RSS temperature and CO2 rate of change one need to multiply the temperature curve with a factor and add an offset. The match of the slopes of the observed CO2 rate of change and the calculated rate of change from the emissions plus the small slope of the natural transient response needed a very small offset to have a perfect match in the slopes. The calculated CO2 slope from the emissions and the observed CO2.dt slope have a small difference, but that is not measurable in the total increase of CO2.

A drawback of the artificial match of the slopes of temperature with the CO2 rate of change is that this also affects the amplitude of the variability as one factor is used to adjust the slopes and the variability. As both are caused by different processes (vegetation is dominant in the variability, but has a zero to negative slope over periods longer than 1-3 years), that leads to a too low amplitude of the variability if the difference in slope angles is large, as is the case for the HadCRUT-SH temperature...

As can be seen in these graphs, both temperature rate of change and CO2 rate of change from humans + natural transient response show the same variability in timing and form. That is clear if we enlarge the graphs for the period 1987-2002, encompassing the largest temperature changes of the whole period, the 1991 Pinatubo eruption and the 1998 super El Nio:

 RSS-emiss-1987-2002

hadcru-T-SH and CO2 derivative compared
                      1987-2002

Fig. 15: RSS and HadCRUT-SH temperature compared to CO2 increase and transient response of natural CO2 + emissions rate of change 1987-2002 [9][11].

As is very clear in this graph, there is an exact match in timing and form between temperature and the transient response of the CO2 rate of change, as was the case in the theoretical calculations. Where there is a discrepancy between the observed and calculated rates of change of CO2, temperature shows the same discrepancy, like the 1991 Pinatubo eruption which increased photosynthesis by scattering incoming sunlight.

It is possible to match the slopes and variability by temperature only or by the effect of human emissions + natural variability.

4. Conclusion:

Which of the two possible solutions is right is quite easy to know, by looking which of the two matches the observations.
The straight forward result:
- The temperature-only match violates all known observations, not at least Henry's law for the solubility of CO2 in seawater, the oxygen balance - the greening of the earth, the 13C/12C ratio, the 14C decline,... Together with the lack of a slope in the derivatives for a transient response from oceans and vegetation to a linear increase in temperature.
- The emissions + natural variability matches all observations. See: http://www.ferdinand-engelbeen.be/klimaat/co2_origin.html

Most of the variability in the rate of change of CO2 is caused by the influence of temperature on vegetation. While the influence on the rate of change seems huge, the net effect is not more than about +/- 1.5 ppmv around the trend and zeroes out after 1-3 years.

Most of the slope in the rate of change of CO2 is caused by human emissions. That is about  110 ppmv from the 120 ppmv over the full 165 years (about 70 from the 80 ppmv over the past 57 years). The remainder is from warming oceans which changes CO2 in the atmosphere with about 16 ppmv/C, per Henry's law, no matter if the exchanges are static or dynamic.

Yearly human emissions quadrupled from over 1 ppmv/year in 1958 to 4.5 ppmv/year in 2013. The same quadrupling happened in the increase rate of the atmospheric CO2 (at average around 50% of human emissions) and in the difference, the net sink rate.
There is not the slightest indication in any direct measurements or proxy that the natural carbon cycle or any part thereof increased to give a similar fourfold increase in exactly the same time span, which was capable to dwarf human emissions..

Conclusion:
Most of the CO2 increase is caused by human emissions.
Most of the variabil
ity is temperature induced variability.
The match between the slopes of temperature and CO2 rate of change is entirely spurious.

All the above reflections of theoretical and observed temperature and CO2 changes were extensively discussed at Anthony Watts' WUWT blog with over 600 comments:
http://wattsupwiththat.com/2015/11/25/about-spurious-correlations-and-causation-of-the-co2-increase-2/

5. References

[1] Why the CO2 increase is man made (part 1)

[2] Engelbeen on why he thinks the CO2 increase is man made (part 2)

[3] Engelbeen on why he thinks the CO2 increase is man made (part 3)

[4] Engelbeen on why he thinks the CO2 increase is man made (part 4)

[5] http://bishophill.squarespace.com/blog/2013/10/21/diary-date-murry-salby.html?currentPage=2#comments

Fourth comment by Paul_K, and further on in that discussion, gives a nice overview of the effect of a transient response of CO2 to temperature. Ignore the warning about the "dangerous" website if you open the referenced image.

[6] http://esrl.noaa.gov/gmd/co2conference/pdfs/tans.pdf
Lecture of Pieter Tans at the festivities of 50 years of Mauna Loa measurements, from slide 11 on.

[7] http://www.sciencemag.org/content/287/5462/2467.short full text free after registration.

[8] http://www.bowdoin.edu/~mbattle/papers_posters_and_talks/BenderGBC2005.pdf

[9] temperature trends of RSS (satellites) and HadCRUT4 (thermometer hut) and the CO2 trend and derivatives were downloaded from Wood for trees.
            CO2 and δ13C trends are derived from the carbon tracker of NOAA: http://www.esrl.noaa.gov/gmd/dv/iadv/
            CO2 emissions until 2008 from:
http://cdiac.ornl.gov/trends/emis/tre_glob.html
            CO2 emissions from 2009 on from: http://www.eia.gov/cfapps/ipdbproject/IEDIndex3.cfm?tid=90&pid=44&aid=8

[10] The spreadsheet can be downloaded from: http://www.ferdinand-engelbeen.be/klimaat/CO2_lags.xlsx

[11] The spreadsheet can be downloaded from:  http://www.ferdinand-engelbeen.be/klimaat/RSS_Had_transient_response.xlsx

[12] http://www.john-daly.com/carbon.htm


Beck's historical CO2 measurements Jaworowski's take on ice cores

On the net: 24 November, 2015.
Last revision: 22 December, 2015.

To the
family home page

To the climate change page

To the CO2 origin page

Zend naar: ferdinand.engelbeen@pandora.be