# How does temperature depend on CO2?

Robert Rohde has produced a very nice animation of global temperatures as a function of CO2 levels in the atmosphere. Of course it is designed for public relations purposes in order to show increasing CO2 causes warming.  He even uses absolute temperatures which are not even directly measured. Here is my version of how temperature anomalies depend on CO2.

Fig 1. Land temperature (GHCN) and Global temperatures (HADCRUT4) plotted as a function of CO2 levels. GHCN-Daily agrees with Berkeley Earth Land temperatures. Normalised to 1961-1990 baseline.

After a rather uncertain temperature rise from pre-industrial (280ppm) temperatures, there is a long period with no net warming between CO2 levels of 300 to 340 ppm, corresponding to the  period ~1939 to ~1980. Warming afterwards continued as expected but then began tailing off towards a logarithmic dependency on CO2.

Many people will often glibly inform you that the CO2 greenhouse effect produces logarithmic radiative forcing, and state that this can easily derived from simple physics. However, few can really explain to you why it should be logarithmic, and it turns out that there is no simple proof as to why it should be. The often quoted formula for radiative forcing:

$S = 5.35 \times \ln{\frac{C}{C0}}$

can be traced back to a paper from 1998 in GRL (Myhre et al)

This formula is in reality a fit to some rather complex line by line radiative transfer calculations by hundreds of vibrational excitation states of CO2 molecules for absorption and re-emission of infrared radiation .  I have perviously described my own calculation of this radiative transfer and how you can fit a logarithmic dependency to it. The physical reason why increasing CO2 apparently produces a logarithmic forcing is that the central lines rapidly get saturated way up into the stratosphere, the strongest of which can then even cause cooling of the surface. Overall net warming is mostly due to strengthening of the weaker peripheral excitation  levels of the 15 micron band.

Fig 2: Calculated IR spectra for 300ppm and 600ppm using Planck spectra. Also shown are the curves for 289K and 220K which roughly corresponds to the Stratosphere. The central peak is cooling the planet because it lies high up in the stratosphere where temperatures are rising.

The net effect produces an apparent ‘logarithmic’ dependency, that I also calculated, and which is very similar to that of Myhre et al. Notice also how 3/4 of the “greenhouse” effect from  CO2  kicks in from zero to 400ppm.

Figure 1: Logarithmic dependence of radiative forcing on CO2 concentration up to 1000 ppm

The effect of increasing CO2 is to raise the effective emission height for 15micron IR radiation photons. The atmosphere thins out with height according to barometric pressure, and eventually the air is so thin that IR photons escape directly to space, thereby releasing energy from the atmosphere. Some IR frequencies can escape directly to space from the surface (the IR window). Others escape from cloud tops or high altitude water vapour and ozone.

The loss of energy from the top of the atmosphere drives convection and evaporation which is the primary heat loss from the surface. This process also drives the temperature lapse rate in the troposphere without which there could be no greenhouse effect. The overall energy balance between incoming solar insolation and the radiative losses to space determines the height of the tropopause and the earth’s  average temperature. A small sudden increase in CO2 will slightly reduce the outgoing radiative loss to space, thereby  creating an energy imbalance. This small energy imbalance is called “radiative forcing”. The surface will consequently warm slightly to compensate, thereby restoring the earth’s  energy balance.

This effect can be estimate from Stefan Boltzman’s law.

$S = \sigma \epsilon T^4$

$DS = 4 \sigma \epsilon T^3 DT$

If you assume T is constant (the answer increases by 1% for 1C if you don’t) then

$DT = \frac{DS}{4 \sigma \epsilon T^3}$

so with T = 288K and $\epsilon \approx 0.6$ and an effective insolation area of the earth of $\pi \times R^2$ this then  gives

$DT \approx 1.6 \times \ln{\frac{C}{C0}} (^\circ C )$

A steeper slope would be expected with net positive feedbacks

Figure 2 shows HadCRUT4.6 and my version of GHCNV3/HadSST3 plotted versus CO2 and compared to a logarithmic Temperature Dependence.

HadCRUT4.6 and 3D-GHCNV3/HadSST3 plotted versus CO2. The orange and purple curves show logarithmic temperature dependencies.

There is still a discrepancy in trends before CO2 reaches ~340ppm but thereafter temperatures follow a logarithmic increase with a scale factor of about 2.5. This implies a climate sensitivity (TCR) of about 1.7C .

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### 31 Responses to How does temperature depend on CO2?

1. gymnosperm says:

“The effect of increasing CO2 is to raise the effective emission height for 15micron IR radiation photons”

Does it?

The P and R branch rotations are seen from 70 km radiating at something like 220K, but the line of sight from 70 km crosses 220K on the tropical lapse curve three times. Which does outer space (or the satellite) register? The first and third crossings would reduce energy to space with increasing altitude; the middle one would increase it.

CO2 upward radiance flatlines above 40 km, suggesting little additional contribution above this. MODTRAN gives 3.4 W/m2 upward radiance above the tropical tropopause.

2. Interesting. I have been in the habit of asking warmistas where the logarithmic coefficient comes from and where else it is found in physics only to get abuse. It is just a curve fit without any detailed understanding, was my suggestion to them. Thanks for providing a reasonable rationale

• Clive Best says:

Yes it is a curve fit that assumes nothing else changes, for example that the lapse rate is fixed.

3. gymnosperm says:

“The central peak is cooling the planet because it lies high up in the stratosphere where temperatures are rising.”

Apologies for being possessed but this post touches on a problem I am struggling with. I cannot replicate the University of Chicago MODTRAN Planck curves and CO2 deviation. Your first graphic above (which has a Spectracalc look and feel) uses ordinary units of radiance. The CO2 deviation nests neatly between 220K and ~300K just as MODTRAN. I was interested in determining more precisely the radiative altitude of the 667.4 fundamental bend.

The MODTRAN “Raw Model Output” states very specifically that “This program computes radiance”. Accordingly, I densified the radiance Planck curves, hoping to better resolve the temperature. Using the standard Planck Radiance:

This is what I got:

A very disappointing result. The CO2 deviation does not nest between 220K and 300K, and the problem is not mere scaling. The MODTRAN output graphic is in units “Intensity W/m2” I queried David Archer about this problem. He kindly sent me his spreadsheet used for the MODTRAN Planck curves. It uses the formula:

His formula appears to apply asymmetric scaling. The red shows difference from standard Planck. Figured this was ad hoc adjustment, but when I replotted using his spreadsheet numbers I got this:

Deja vu all over again. No nesting.

Whether this is an embarrassing personal problem or it calls to question the validity of blackbody curves for temperature, the 667.4 peak has an intensity that bedevils Planck and Stephan-Boltzmann analysis.

The highest published column emissivity for CO2 I can find is .2. Just a lousy blackbody.

4. Nick Stokes says:

“can be traced back to a paper from 1998 in GRL (Myhre et al)”
That is a calculation of the coefficient. But as to why the dependence of equilibrium temperature on CO2 should be logarithmic, it was certainly the firm view of Arrhenius. Here he is spelling it out in his 1908 book:
“If the quantity of carbonic acid in the air should sink to one-half its present percentage, the temperature would fall by about 4°C; a diminution to one-quarter would reduce the temperature by 8°C. On the other hand, any doubling of the percentage of carbon dioxide in the air would raise the temperature of the earth’s surface by 4°C; and if the carbon dioxide were increased fourfold, the temperature would rise by 8°C.”

So where does he get it from? At the end of his 1896 paper p 267
“Thus if the quantity of carbonic acid increased in geometric progression, the augmentation of the temperature will increase nearly in arithmetic progression”
He gets it empirically from the calculations he made (Table VII)

• Clive Best says:

This is fascinating !
I found Arrhenius’s original paper. He used measurements by Langley at the surface of the ‘IR radiation’ from the full moon to measure IR absorption coefficients by the earth’s atmosphere. The amount of CO2 and H2O traversed was estimated from the incident angle. The absorption coefficients basically assume a log dependence.

5. Hans Erren says:

Nick, Arrhenius’ calculations were based on infrared spectral observations in the Allegeny Observatory in Pittsburgh that stopped well before the CO2 infrared absorption band.
The logarithmic nature of bulk infrared absorption can only be seen in lab experiments starting from 1900 when sufficient CO2 pressure could be generated that equaled atmosperic optical depth. As shown in my presentation in Stockholm in 2006. http://www.kolumbus.fi/boris.winterhalter/KTH/HanErr.pdf

6. Hans Erren says:

John Koch, Beiträge zur Kenntnis der Wärmeabsorption in Kohlensäure., Öfversigt af Kongl. Vetenskaps-Akademiens Förhandlinger, 1901. N:o 6 p 475-488

7. dpy6629 says:

Clive, One could argue that you should use Global temperature (including over oceans) to calculate TCR. What does your method produce in this case? It should be close to Nic Lewis’ estimate of TCR because its based on the same data.

• Clive Best says:

The value is based on using Land+Ocean (HADCRUT4 or variants thereof)

I respect Nic Lewis’ work , but I just get a slightly higher figure than he does.

My estimates of TCR and ECS are described in The strange case of TCR and ECS

My estimate for ECS is 2.3C.

This is about half the value Arrhenius predicted in 1896.

The world will be absolutely fine if CO2 eventually doubles. What happens to human civilisation though is another matter. The only long term hope is some breakthrough in Nuclear Fusion.

Try reading Energy & Civilization A History by Vaclav Smil !

• Ronald Chappell says:

Please don’t be so pessimistic, Clive. We already know and have proven the liquid salt breeder reactor in research phase and can make a nuclear fueled power source within the known state of the art and when the Chinese finish the development which they have seriously committed to, we backward Western nations will be buying our reactors from them. (powered with virtually unlimited Thorium)

8. A C Osborn says:

Clive why do you think that 100% of the warming is due to CO2?
If not what %age is it.

9. Hans Erren says:

Clive could you redo the graphs using a log scale for CO2?

• Clive Best says:

Here is a 5y averaged global temperature anomaly plotted on a log(CO2) scale. The agreement is really not that good before ~1990.

10. Damien says:

Hi, i don’t understand the cooling effect of the central band. Can you explain this more clearly ? Thanks !

• Clive Best says:

This is because the temperature in the stratosphere increases with height so consequently emissions to space from the strongest central lines will increases with rising CO2 levels, as the emission height increase. This is the opposite effect of what happens in the troposphere.

You can see the central spike at a higher temperature in this Nimbus IR spectrum.

• Hans Erren says:

Clive, one thing that puzzles me already a few years is the the fact that in winter inversion temperature profiles, the co2 band is an emission band and increasing co2 will kead to a higher emission of the atmosphere and so a cooling effect. So we would expect over antarctica siberia and canade to see a cooling trend in winter as a result of increasing co2

• Clive Best says:

I agree. Winter inside the Arctic or the Antarctic circle essentially reaches zero insolation, and the stratosphere drops down to the surface. That means that that those CO2 emission levels high up are actually warmer’ than the surface .

In summer though the annual meridional temperature gradient reduces more with increased CO2.

• Hans Erren says:

But why is the february temperature anomaly the highest over Siberia and Alaska, what is the warming mechanism?

• What is the warming mechanism?

Well, it is obviously not CO2. It must be something that is easily overwhelming the cooling effect of CO2 there. That is if you believe in the CO2 knob.
Otherwise, who knows? Some combination of known and unknown unknowns.

• Damien says:

Okay ! But does not the fact that warming leads to elevating the altitude of the tropopause compensate for this effect ?

• Hans Erren says:

The polar tropopause is separated from the lower lattitude tropopause

11. Hans Erren says:

(Apologies for the typos)

– Fixed – Clive !

12. Steve Crow says:

I got results similar to yours with the HITRAN data base, from 550 to 750 cm-1. That range includes 16,017 of the 22,666 CO2 absorption lines in the 2017 version of HITRAN. I don’t know how to attach plots of my results, and maybe you can help. I found a logarithmic relation between DT (deg C) and c (ppmv) for c between about 10 and 1000 ppmv, with saturation above 2000 ppmv at about 12 deg C. The centers of many absorption lines are saturated at low values of c, but the skirts of the line profiles rise to absorb more CO2. I used Lorentzian absorption profiles, appropriate for pressure broadening.

My calculations for Mauna Loa CO2 concentration data imply a temperature anomaly on 0.12 deg C in 1958 and 0.37 deg C in 2017, an increase of 0.25 deg C. No distinction is made between natural and man-made CO2 and none is needed, since the causal contribution of CO2 to the observed temperature increase is small. I share your curiosity about the actual cause of the temperature in increases in the arctic and Siberia. Happily, the temperature increase in Boulder, CO has been negligible for the past 120 years 🙂 Steve Crow

• Clive Best says:

That sounds about right. Here is my temperature GHE for CO2 alone as a function of CO2 concentration.

To place an on-line image in a comment, you just need to add the URL on a line by itself in the comment and it will appear. I normally add a blank line on either side.

• What I don’t get is that you place trust in HITRAN, which has been created by scientists that evidently know what they are doing, yet you then draw conclusions from HITRAN which are at odds with the same scientists that should still know what they are doing.

• roberto says:

Hi Clive,
there is no validated peer review document that demonstrates CO2 influences the JS and the climate change anthropic! climate and CO2 are ignored.
it is the persistence of the Jet Stream in the time that regulates the temperatures over time. In Italy, in this winter, Russian Arctic retrograde flow of the Jet Stream, has favored snow to the south of Italy even in lowland areas (city ??on the sea) and higher temperatures in northern Italy
and little snow! (Po Valley).

13. Anton Beijer says:

The increase in temperature in Siberia as well as the low temperatures in Northern America were a result of the severe polar jet meandering in feb. with a abnormal North polar jet stream position abeam Europe and Siberia.

The polar Jet prevented cold influx from the North Pole into Europe and Siberia. In North America the reverse happened: Polar jet there followed a very Southern path allowing cold Arctic air tp stream South over Canada up to the middle part of the USA cutting off airflow from the Pacific into North America and blocking warm air from the South and West.