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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48 Responses to How does temperature depend on CO2?

  1. Pingback: How does temperature depend on CO2? – Climate Collections

  2. 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.

  3. 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

  4. 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.

    • Frank says:

      Gymnosperm and others: To make you calculations come out correctly, you need to consistently use units of kilograms, meters and seconds. Unfortunately, spectroscopists like to use units of cm^-1, instead of frequency. The exponential term should be exp(hv/kT) where v is the frequency. However, Archer is using v for the wavenumber (cm^-1). If you carefully sort through Archer’s work, you’ll probably find the 10^8 term comes from using cm instead of m, possibly in the speed of light as well as wavenumber.

      The factor of Pi and perhaps some factors of 10 come from confusion about the differences in unit of power transmitted by radiation: flux (W/m2) and intensity (W/sr/m2) and spectral intensity (W/sr/m2-nm, W/sr/m2-um, or W/sr/m3) and spectral flux (W/m2-nm, W/m2-um, or W/m3). Planck’s Law affords spectral intensity, and there many factors of 10 involved between the units involving nm, um or m. To get intensity, one needs to integrate spectral intensity over a range of wavelengths.

      Finally Planck’s law is all about radiation emitted in all directions – over a solid angle of 4Pi. Flux is all about that component of radiation traveling in all directions that is moving perpendicular to a surface. In the atmosphere, we use what is called the “two-stream” approach to extract the vector component of the radiative flux traveling in the +z direction, from that vector component of the flux traveling in the -z direction, and from the vector components traveling in the x and y directions that cancel. So we take the intensity (W/sr/m2) traveling towards an upward hemisphere (2Pi stenradians) and integrate the +z component over that hemisphere and are left with a factor of Pi. IIRC, an intensity of 10 W/sr/m2 becomes a flux of 5/Pi in the +z direction and 5/Pi in the -z direction. And a flux of 390 W/m2 traveling upward from the surface has an intensity of 390*Pi W/sr/m2 traveling in all directions (towards a hemisphere, not a plane).

      Note: I haven’t spent serious time interconverting W/m2 and W/sr/m2, so it is possible some details above are incorrect. But I’m confident that your extra factor of Pi arises from this interconversion.

    • Frank says:

      gymnosperm: The apparent contradictions you find in David Archer’s work are caused by misunderstand about the units being used. Reciprocal centimeters (cm-1) are proportional to frequency, but not expressed in the standard units (kg, m, s) for frequency (s-1). And sometimes when people work with cm-1, they use cm/s (not m/s) for the speed of light.

      Planck’s Law gives you the spectral intensity (technically spectral radiance) of radiation, which may be reported in units of W/sr/m2/nm, W/sr/m2/um, W/sr/m3, or W/sr/m2/Hz. This number needs to be integrated over a range of wavelengths or frequencies to obtain the total intensity (radiance, W/sr/m2) being emitted by a source.

      Finally, we think in terms of the flux (W/m2) crossing the TOA orbeing emitted or absorbed by the surface. The extra factor of Pi arises from converting intensity/radiance to flux. The surface of the Earth emits an average flux of 390 W/m2 perpendicular to the surface, but it also radiates 390*Pi W/sr/m2 in all directions towards a hemisphere. The GHG’s in the atmosphere, which radiate in all directions, so we apply a “two-stream” approach to the flux. The vector component in the +z direction emitted towards an upward hemisphere is the upward flux which becomes OLR at the TOA, and the vector component in the -z direction emitted towards a downward hemisphere becomes the downward flux called DLR, and the vector components of the flux in horizontal directions are irrelevant from a heat transfer perspective.

    • Frank says:

      The area under the Blackbody curve for 300 K should be 459 W/m2 or 459*Pi W/sr/m2. Above I noted that spectral radiance can be reported in units of W/sr/m2/nm, W/sr/m2/um, W/sr/m3, or W/sr/m2/Hz, but I should have included W/sr/m2/cm-1.

      At the Modtran website when using wavelength, the units on the vertical axis are W/m2-um when using wavelength. If approximated as a triangle the base is about 30 um wide and the peak is 32 W/m2-um. Using cm-1 (which are proportional to energy), the units on the vertical axis are W/m2-cm-1, and the peak value is almost 0.5 W/m2-cm-1, the same as on your graph. The base of the “triangle” is about 2000 cm-1 wide, both of which agree with what we expect for a blackbody at 300 K.

      The units on the left hand axis of your Figure should be W/sr/m3 if you entered all values in standard units, but I can’t convince myself this is right. It doesn’t make sense to put these units on the y-axis and cm-1 on the x-axis. Working with units of spectral radiance is a pain.

  5. 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.

  6. 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.

  7. 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

  8. 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)

        • Clive Best says:

          I hope you’re right.

          • Bindidon says:

            I’m afraid he is not.

            Thorium is indeed virtually unlimited, but it is only a fertile element, and no fissile one.

            You have to breed your Th232 up to U233. But the blind-alley here is that near U233, the U232 isotope is produced as well.

            And this element not only is highly toxic, is neither fertile nor fissile, but above all it also slows down the U233 chain reactions.

            As far as I’m informed, you can’t separate the two isotopes at reasonable cost.

            This is known since decades.

  9. A C Osborn says:

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

      • JPinBalt says:

        What about oxygen? For every CO2 molecule we put in the atmosphere we reduce an equivalent amount of O2 or O3 in the atmosphere. Oxygen has absorption bands too in the electromagnetic spectrum. If CO2 is rising, O2 and O3 are falling, and observational data shows this. Would not oxygen depletion have an offsetting effect on temperature?

        • Clive Best says:

          That’s correct, but only Ozone has absorption bands within the IR spectrum for the earth at 282K. Although O2 is decreasing molecule for molecule in step with increasing CO2, the percentage change in O3 GHE is still tiny. This is simply because there is 21% O2 and only 0.04% CO2. So a doubling CO2 would reduce oxygen in the atmosphere by only 0.1%. Likewise the decrease in O3 GHE is just 0.1%

  10. Hans Erren says:

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

  11. 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 ?

  12. Hans Erren says:

    (Apologies for the typos)

    – Fixed – Clive !

  13. 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).

  14. 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.

  15. Ulrike Pielmeier says:

    Clive, your blogs have given me a lot of inspiration – thank you very much for putting this much work into it.
    I have a few questions, which have puzzled me a while, and I wonder, if you could provide me and perhaps some others with some insight:
    1. Why should any effect of CO2 on temperature matter at all at the top of the troposphere, where pressure is very very low? I think I read that at least in the troposphere the gases are well-mixed, and that the concentration of CO2 remains stable with increasing height. That means, there are really very few CO2 molecules left in 12-15 km altitude?
    2. Why does the stratosphere consistently cool? According to UAH Upper Air Temperatures in 17 km, the temperature records since 1979 show a decrease of roughly 1.2 degrees celsius. In one of your blogs it is written that cooling is due to less water vapour, but this cannot be the cause, since the water vapour content of air is temperature dependent, so it is the other way around. However, the question remains why this cooling takes place?
    3. None of your models (or others) seem to mention ozone. Ozone is produced in the stratosphere and isn’t it much more likely to be the primary driver of the emission height for escaping protons, as you use to call it? Ozone production/ozone layer thickness as far as I know also varies and seems to be subject to (natural?) cycles. I cannot help but wonder why CO2 in high altitudes should have a major impact (where it is almost not existent) and O3 does not?
    Thank you so much in advance.


    • Clive Best says:

      Thanks Ulrike.

      You have some great questions. I am not sure I can answer all of them properly but here goes:

      1. The ‘Top of the atmosphere’ argument is just a thought experiment. CO2 does not warm anything. The sun alone warms the earth’s surface which radiates heat upwards. The greenhouse effect relies on temperature falling with height (lapse rate). Temperature is the mean kinetic energy of air molecules. A tiny amount of those molecules are CO2 which can be excited into vibrational energy states then emitting 15 micron photons. Those photons excite other CO2 molecules above them at lower temperatures, but there are fewer of them as density falls with height. Eventually the air thins enough for 15 micro photons to escape to space. This is quite complicated so the only way to calculate the overall effect is to estimate the imbalance between insolation and outgoing IR with increasing CO2 ‘at the top of the atmosphere’ !

      2. I have been on a learning curve as well so some of my earlier blogs could well be wrong. The stratosphere cools IMHO simply because the strongest vibrational lines in the CO2 spectrum are already saturated way up in the stratosphere and temperature rises with height mainly due to ozone production by sunlight. This means that more energy is lost to space from the stratosphere i.e. increasing CO2 causes cooling in the stratosphere.

      3. Ozone is produced also near the ground both naturally and due to human pollution and it is a greenhouse gas. You can see an ozone dip in IR spectra from satellites. However it is probably less important because its lifetime in the atmosphere is very short. So yes the net effect of us producing more ozone near the surface has a net warming effect but this quickly disappears once we reduce for example NO2 emissions.

  16. i’m sure everyone knows more about the chemistry of u232 separation than I do, however many have looked at this and seem to believe that It can be done. I do know that something along the line of a gen 4 reactor must be solved. maybe the fast reactor burning up our huge store of ‘waste’ needs to be done first. I leave this to the nuclear reactor physicists and engineers.

    • Mike Ellwood says:

      Yes…unfortunately nuclear fusion is too far off to provide any realistic solutions in a useful timeframe, but we know that MSR’s (molten salt reactors) can work because it’s been done before. The basic science is understood. There are significant challenges, but those are essentially engineering and chemistry, and are not insurmountable.

      Maybe we can’t do thorium right away (although Kirk Thorensen might disagree), but we could make a start using uranium, and as you suggest, start adding some of our nuclear “waste” into the mix. And if MSR breeders could be made to work, then the problem, even if it did not go away, would be much mitigated.

      A good article (showing pros and cons) on MSRs:

      And an article showing how it’s now feasible to get uranium from seawater (still in early stages…):

  17. climategrog says:

    Hi Clive, interesting article and some solid looking work.

    I’m interested in your temp vs CO2 plots eg the log plot here:

    Firstly I would point out that HadCRUFT4 is not land temps as you label the graph but a physically non meaningful land + sea index. You can not average temperature, certainly across different media like this. ( Land changes roughly twice as quick as sea to the same “forcing”.) That this averaging is frequently done in climatology just shows the lack of rigour and understanding of basic physics of many involved, despite constant claims that it is all founded in “basic physics”.

    When doing physical calculations you would need to look at either land or sea and interpret the result for that part of the earth’s surface, not use a land+sea chimera. Since it is oceans which really control the planet, I would suggest SST.

    Now the big take away from this graph, for me, is that it really blows away the suggestion that GHE accounts for the majority of the variability. The mid-century dip clearly does not work neither does the early 20th c. rise. In short, it is only the post 1970 rise which roughly follows the idea and that is because it is basically monotonic.

    Aligning two monotonic changes is not good evidence of causation. What you need is a decent level of correlation when there are changes in both directions and that is where the suggested relationship breaks down.

    What this graph would seem to show is that are other, probably natural phenomena, which account for a large proportion of temperature change. That would imply a lower sensitivity.

    Regards, Greg.

  18. Dan Pangburn says:

    Considering only three factors; an approximation of the net effect of ocean cycles, the time integral of SSN anomalies, and TPW (which is the absolute water vapor content of the atmosphere) has resulted in a 98+% match to measured average global temperature 1895 thru 2018.

    Increased absorption by CO2 at sea level is effectively balanced by increased emission at high altitude with the result being that CO2 has little, if any, effect on climate.

  19. Dan Pangburn says:


  20. Jim Phelps says:


    To follow are three points regarding “How does temperature depend on CO2?”

    1. Stefan Boltzman’s law is based on a flat surface, but earth is a sphere. Would your results be different doing spherical calculations? I located this paper advocating the spherical alternative complete with the math. THE STEFAN-BOLTZMANN LAW, a simplified derivation, by Miles Mathis (
    2. The majority of greenhouse gas is water vapor; does the same logarithmic influence apply to H2O as it does to CO2? Is this worthy of a sequel, “How does temperature depend on H2O?”

    3. In my less-sophisticated work on school operations, I convert all variables to standard scores and compute the R2 (explained variance) for each input variable—the effect size. Every school has a different location on the normal curve for each variable. Using the cumulative normal curve representing the amount of moving up or down based on the change of inputs, with each student outcome represented by a separate equation with input terms, and an equation representing the cost of changing the inputs, (a set of simultaneous equations) I calculate an optimal solution for every individual school—in contrast to the average school. This works with nonlinear asymptotic functions based on the normal curve.

    My point, is it possible to apply the same idea in order to obtain results for individual global locations? (When I go to the doctor, I am more interested in a treatment applied specifically to me and not to the average person.) This provides the opportunity to add localized factors influencing temperature. The results would be a more comprehensive explanation of how the total system works. (E.g., is CO2, H2O, albedo, altitude, currents, etc. the same everywhere? If not, what is their specific influence?) (E.g., your examples for England and Australia, also Steve Crow in Boulder, CO, USA, a mile high.) Would it work with logarithmic curves?

    This point is made in a reply to your paper:


    I am 84, my last math class was over 60 years ago and I am self-taught in science, mostly physics—my bed-time reading. I have a general understanding but not capable of doing calculations. Your work is the most sophisticated and objective of all that I read.

    Thanks, Jim Phelps

  21. Frank says:

    Clive: The problem with your graph and Rodhe’s is that global warming is driven by increasing radiative forcing, not just increase increase in CO2. The radiative forcing from CO2 varies approximately with the log of the CO2, so your x-axis should be logarithmic. I’ve gotten the radiative forcing data from AR5, removed the volcanic forcing and found a nice linear relationship between forcing and global warming with a slope of 0.4 K/(W/m2). That is a TCR of 1.4 K assuming F_2x of 3.45 W/m2.

    Volcanic forcing needs to be omitted for the following reason: If you impose an instantaneous radiative imbalance of +1 W/m2 and all the additional power remained in the atmosphere and a 50 m mixed layer of ocean, it is simple to calculate that the temperature will start rising at an initial rate of 0.2 K/yr. However, as the planet warms it will begin to radiate more LWR to space (and perhaps reflect more or less SWR too). If climate sensitivity is 3.6, 1.8,or 1.2 K/doubling, the change in net radiative cooling to space will be -1, -2 or -3 W/m2/K and the warming at equilibrium from a +1 W/m2 will be 1, 0.5, or 0.33 K. Consider the case where equilibrium warming is 1 K. 2.5 years of 0.2 K/yr would get us half way to equilibrium warming, at which point the radiative imbalance at the TOA would be down to 0.5 W/m2 and the rate of warming down to 0.1 K/yr. So it takes about a decade for the “transient” response of the mixed layer to approach equilibrium. And somewhat longer when we recognize that some heat is escaping to the deeper ocean below the mixed layer. If climate sensitivity were 1.8 or 1.2 K/doubling (-2 or -3 W/m2/K), the process of approaching approaching an equilibrium response in the mixed layer and atmosphere will happen 2-fold or 3-fold faster, because the initial warming rate will still be 0.2 K/yr, but the equilibrium response will be 2-fold or 3-fold smaller. When a volcano imposes a large negative forcing that disappears in a few years, it is completely impossible GMST to respond as one might expect from a 1% pa experiment.

    So, by ignoring volcanic forcing, I get a fairly straight line relationship between forcing and warming with the years immediately after the volcano somewhat too cool (about like years with strong La Ninas). And hidden behind that straight line is a realty that the warming esponse to a small increase in forcing between two years is only about 1/5 complete in that year.

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