The relationship of CO2 to temperature

Is temperature a function of CO2 or is CO2 a function of temperature i.e.

Is T=f(CO2) or CO2=f(T) ?

The answer according to Richard Betts is both, because increasing CO2 warms the planet but levels in the atmosphere also respond to natural ENSO cycles. However for the last 50 million years until now, CO2 levels have only ever reacted to changes in temperature rather than being the primary driver of temperature. Glacial cycles are driven by regular variations in the earth’s orbit around the sun. The most important of which are changes to the tilt of the  earth’s rotation axis to its orbital plane determining the distribution of solar energy with latitude and with season. During a glacial cycle CO2 levels rise in synch with temperature but fall out of synch with reducing temperatures, never exceeding 290ppm. The Biosphere seems to regulate maximum CO2 levels at ~280ppm for as long as possible into the next glaciation. Eventually spreading ice sheets destroy this Biosphere balance and CO2 levels then fall rapidly, reaching as low as 190ppm at LGM.

Details of the Eemian termination. Note how CO2 rises in synch with temperature, but then never exceeds 280ppm even though the Eemian was at least 2C warmer than the Holocene. CO2 then remains at 280ppm while  temperatures fall 4C.

Today though we have a totally different situation. Human emissions of CO2 since 1900 are artificially increasing CO2 levels during an otherwise fairly cool interglacial. CO2 has rapidly increased beyond 280 ppm ahead of any associated increase in  temperature. So now we are studying the T = f(CO2) dependence. Yet such a situation has only occurred once before, some 55 million years ago (PETM), when the earth’s climate was already far warmer than today.

Artificially increasing CO2 must reduce slightly the IR radiation escaping to space from the atmosphere. This results in a sudden energy imbalance causing the surface to warm slightly in compensation, thereby restoring energy balance. The reduction in outgoing IR energy with more CO2 is because the effective height of emission rises to a colder level. All layers in the atmosphere, including the surface, are in local thermodynamic equilibrium at the appropriate lapse rate temperature. Of course if the surface warms up a bit then there will also be a bit more evaporation and more clouds because 70% of the earth is covered by oceans. This is where it gets complicated and no-one really knows how much it will warm if we double CO2. Climate models may claim to take all these feedbacks into account but at the end of the day they are just guesses, which is also why there is such a large spread in resultant climate sensitivity.

During Glacial cycles climate change is induced by changes in orbital insolation and CO2 comes along for the ride as a feedback. Interglacials begin with a burst of very fast warming. Initially increases in CO2 are due to outgassing from warmer oceans. However the biosphere then flourishes in this new warmer and wetter climate. This seems to limit the increase in CO2 such that levels never exceed 280 – 300 ppm independent of how warm the interglacial gets. There is not a simple one to one relation between temperature and CO2. CO2 ne f(T) when the climate is driven by orbital cycles. Interestingly though 280ppm also happens to be the peak for the atmosphere to radiate heat to space for the current climate.

Change in atmospheric OLR with CO2 concentration for a fixed surface temperature of 288K.

280ppm is optimised for radiative cooling of the atmosphere by CO2 in today’s climate. This cooling then drives convection, generates the lapse rate and bootstraps the greenhouse effect.

To understand better what happens if CO2 suddenly increases while everything else  remains constant we need to look at PETM 55 million years ago. The climate conditions then were radically different than those of today.  CO2 levels were 1000ppm and temperatures  8-10C warmer than today. The Arctic ocean was isolated and warm. The Himalayas had yet to form, and Antarctica was joined to South America and still ice free. The Atlantic ocean was expanding and North America was about to split from Eurasia.

Continents as of PETM

Up to 6000 GTC was emitted into the atmosphere during a short period of time at PETM.  For comparison, human emissions to date are 500 GTC and if we were to burn all known reserves of fossil fuels they would become 2000GT. The source of the carbon for  PETM is not known but may be connected to volcanism associated with the opening of the Atlantic ocean

This is a reconstruction from ocean core proxy measurements of PETM (taken from the book ‘Paleoclimate’ by Michael Bender).

Temperatures rose by about 5C and both CO2 levels and temperatures recovered within about 100,000 years. Does this tell us what will happen if humans double CO2 levels ? The main difference today is that the natural climate is much colder than the Eocene with regular 100ky glacial cycles. The temperature falls by over 5C between an interglacial as now and a glacial minimum. The likely effect in the Holocene would be that an anthropogenic warming of  ~4C would be reduced back to zero as the earth’s tilt reduces, allowing the earth to  skip the next ice age, which I estimate will start in 15000 years time.

Compare the Anglian and Holocene interglacials

Orbital parameters today are very similar to the Anglian interglacial with low eccentricity suppressing precession of the equinoxes. For this reason the Holocene would naturally have lasted longer than normal. However a minimum obliquity will occur in 12000 years time, and this always heralds a new glaciation.

Near term global warming is not to be welcomed and will surely have short term negative effects. However, such an AGW spike peaking in 2100 would have a silver lining by making the next glacial cycle habitable for North Americans and Europeans !

About Clive Best

PhD High Energy Physics Worked at CERN, Rutherford Lab, JET, JRC, OSVision
This entry was posted in AGW, Ice Ages and tagged , . Bookmark the permalink.

17 Responses to The relationship of CO2 to temperature

  1. Pingback: The relationship of CO2 to temperature – Climate Collections

  2. petroalbion says:

    Remembering how the apparent correlation between CO2 and temperature during the last 800k years encouraged the belief that CO2 was a driver for the ice ages, and how this view still hangs on in some circles, how certain are we that the amount of CO2 released is as it is because it is derived from the temperature increase via the AGW formula. In other words do we know absolutely which came first? Given that there are many examples of CO2 responding to changes in temperature and few of the reverse?

    • Clive Best says:

      CO2 is clearly not the driver for ice ages. It is simply a feedback that aids interglacial warming to a limited extent. A far greater feedback is the reduced albedo from melting ice and enhanced H2O greenhouse effect as the climate warms.

      • FSR says:

        Dear Clive,

        From ice-cores you cannot say (given the uncertainties) whether the temperature drops first or the CO2 so could you tell me how you can state that it has always been the temperature that drove the CO2 changes?

        Just to be clear with the question above I am not saying that the CO2 was the driver of ice-ages. I agree that the ice ages were not triggered by the CO2 decrease but the CO2 acted as a positive feedback, which further decrease the temperature, which allowed to increase the ice cover (hence the albedo).

        Furthermore, I would like to point out that the “slight” increase/decrease in temperature due to CO2 can trigger important feedbacks related to H2O, albedo, circulation etc … do you agree?


        Francesco S.R. Pausata

        • Clive Best says:

          I agree that CO2 acted as a positive feedback, likewise H2O. However, the solubility of CO2 in water follows Henris’s law which is temperature dependent. Decreasing insolation at northern latitudes cools the oceans which absorb more CO2 which reduces the greenhouse effect as a feedback. The opposite happens when insolation increases. So CO2 is a feedback.

          It is only now that Humans have increased CO2 before a temperature rise. This is a different phenomenon. Despite this, I don’t think we will avoid another brutal ice age due within 15,000 years.

  3. Steve Crow says:

    I like your empirical correlations between CO2 and T, but it is perfectly possible to derive a relation between CO2 and T for a “simple” model of the atmosphere. The reason for the quotation marks around simple is that around 11,000 spectral lines are needed to account for the absorption of heat by CO2 in the relevant wavelength bands. The calculation can be done as a giant spreadsheet but is nonetheless straightforward. One result is that the increase of CO2, as measured at Mauna Loa since 1958, has caused a temperature increase of 0.25 deg C, much smaller than the observed increase of around 1 deg C. Something else is responsible for the bulk of the warming.

  4. Clive Best says:

    It’s straightforward if you assume a standard atmosphere like MODTRAN does, or my own attempt
    at a line by line calculation see:

    These methods give a value for ECS of around 1.1 C but ignore feedbacks which are mainly H2O and clouds. Your guess is as good as mine.

  5. oldbrew says:

    Is temperature a function of CO2 or is CO2 a function of temperature?

    Neither. Water vapour is by far the main radiative gas in the atmosphere, not CO2. Whether one thinks radiative gases are the only, or most, relevant factor in global temperature variation is another matter.

  6. Steve Crow says:

    I also used the HITRAN CO2 data base with Lambertian line shapes. I see on looking back that I used about 16,000 lines of the 22,666-line data base. I could have cut that down by culling weak lines, the that would have been harder than simply including them.

    My atmospheric model was simple, but the absorption physics was sophisticated. Sunlight comes down without attenuation, is absorbed and reradiated by the ground as albedo plus IR radiation. The IR is absorbed and reradiated isotropically by the atmospheric CO2. Once the spherical spreading is integrated out, the propagation physics reduces to two differential equations on altitude, both having sums over all those lines. Ground temperature is set when the power flux of sunlight at the top of the atmosphere less the albedo equals the outgoing IR power flux.

    The result is a 0.25 deg C increase in temperature for the Mauna Loa CO2 increase since 1958. The increase was from 280 to 400 ppmv during that interval. The temperature increases only logarithmically at higher CO2 levels. A doubling of CO2 from 400 to 800 ppmv causes an additional temperature rise of 0.67 deg C, and further doublings produce about the same increase.

    I am quite confident in the physics of my approach and think the actual higher temperature increases since 1958 are caused by phenomena other than atmospheric CO2.

    • Ron Graf says:

      Diminishing ice albedo certainly accounts for some of the warming. Black carbon aerosols may be amplifying effect. But other anthropogenic aerosols (pollution) increase atmospheric radiative emission, having a cooling effect.

      We also need to always keep in mind that the temperature record is continually being adjusted to account for measurement biases. Whether biases are being removed or added depends on who you talk to.

      WRT “the next glacial cycle habitable for North Americans and Europeans,” either humans will be in technological control of the Earth’s climate with 200 years or humanity would have failed. Within a 1000 years a successful continuation of technological evolution would place humanity or its successor treking the stars, fondly remembering when those who were primitive Earth captives. Stephan Hawking may have been good at math but lacked a wider imagination.

  7. Hans Erren says:

    That is the CO2 thermometer, discoverd by Jarl Ahlbeck in 2001

  8. Hans Erren says:

    Clive, please have a look at this calculation of SRES A1B and A1T scenarios under the first law of diffusion and 1.33 transient climate sensitivity.

  9. Clive Best says:

    very nice. If the airborne fraction decreases with emissions according to the law of diffusion, then it must stabilise.

  10. Ulrike Pielmeier says:

    Hi Clive, Could you provide the source of your figure “Change in atmospheric OLR with CO2 concentration for a fixed surface temperature of 288K” – or the set of equations used – for reproducing it?

  11. Ulrike Pielmeier says:

    thank you.
    Dr. Rex Fleming (a former director within NOAA; his resume from his own website: has published a review paper last year, which I read. He writes:
    “The important equations for radiative transfer are Planck’s equation for the intensity of radiation, and the integration of the Schwarzschild equation for net diffuse radiation. One can see (Houghton 1985) and (Liou 2002) for details.”..
    “The Schwarzschild eq. solution using the Liou notation above and equations of Houghton is:
    F = – ? B (?, T) (d?*) / du) du + ? B (?, T) (d?*) / du) du; the optical depth (d?*) is due to the radiation being diffuse rather than a parallel beam.
    In the first integral, the integration proceeds downward along the optical path from the reference level (Z) with optical path u downward to the surface where the optical path = 0. In the second integral, the integration proceeds upward from the reference level (Z) with optical path u upward to the top of the atmosphere where the total optical path is u1. Both integrals are positive, because of the convention that the path length is measured positive down and then positive up respectively. The net flux at a level is the upward flux at the bottom of a layer minus the downward flux at the top of a layer.”
    He then goes step by step through the exact calculations for different bands of CO2 and summarises his results:
    “One can summarize these calculations as follows: whatever the “climate-change regime”, whatever surface heat from the Sun on any given day within that regime, that heat is fully absorbed and fully vertically redistributed throughout the troposphere – there is no propensity for CO2 to store heat in a systematic way over time to produce a climate-change effect (as defined in the introduction).
    Why does the integrated effect of CO2 have so little effect on the total temperature profile? The reason is that the Planck function change with height (temperature) is very strong in reducing the intensity of those relatively few lines with large absorption coefficients. Another reason is that the longwave radiation is diffuse which depletes the intensity rapidly over distance. The diffuse nature of the radiation also leads to the fact that the net radiation for a given level (that sent upward at the bottom of a layer, minus that sent downward at the top of a layer) further reduces the adsorbed CO2 radiation intensity.
    Other so-called “greenhouse gases” (some with larger absorption coefficients, but all with significantly less concentration) have their intensity quickly transferred upward and depleted by the same strong Planck function intensity change that applies to CO2 and H2O.”

    You will find the link to the paper here, the relevant section regarding CO2’s greenhouse effect (including the Schwarzschild equation) is in Section 5. The review has been published in a peer-review journal last year. It would be nice if you could comment 🙂 All the assumptions and numbers he uses are included in the paper.

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