CO2 Global Warming

It is often said that the science behind CO2 enhanced global warming is simple  and settled. However, calculating the top of the atmosphere ‘radiative forcing’ of CO2 is really not that straightforward at all. First you need to assume an initial temperature profile in the atmosphere and then calculate the net upward transfer of IR photons to space from the surface. This procedure is called  line-by line radiative transfer. The term ‘line’  refers to hundreds of  quantum energy transition levels for vibration and rotation states of CO2 molecules.

CO2 absorbs and emits photons mainly in the 15 micron band. Chemists have measured the absorption coefficients (cross-sections) for each of the individual lines, and these are available in the Hitran database. One must assume a standard atmospheric temperature and density profile and then loop over all lines to calculate the net outgoing radiation flux from the top of the atmosphere.  Next you repeat the calculation many times for small increases of CO2 in the atmosphere. The radiative forcing DS is then equal to the reduction in outgoing IR for a given CO2 concentration. This energy imbalance is the difference between the incoming solar radiation and outgoing IR radiation. This imbalance is assumed to be offset mainly by the surface warming up a little (DT) so as to increase IR flux to compensate. So it is really the sun that causes global warming not CO2 at all. All that increasing CO2 does  is to adjust the radiative profile with altitude.

Once you have done all that, you find that there is an approximately  logarithmic fall-off of forcing DS with CO2 concentrations. The main reason for this is that most of the strongest lines are already saturated up to the tropopause where any increased concentration (height) makes no change to the outgoing flux because temperature remains constant with height. In fact the central line is saturated so far up into the stratosphere that it actually cools the planet with increasing concentrations because temperatures rise higher in the stratosphere. You can see that in both the calculated and measured spectra as shown below.

Fig 7: Calculated IR spectra for 300ppm and 600ppm using Planck spectra. Also shown are the curves for 289K and 220K

My calculation of  IR spectra for 300ppm and 600ppm using Planck spectra for CO2 alone. Also shown are the curves for 289K and 220K

Spectrum taken from Nimbus ii over the equator. The fine structure in the CO2 absorption bite shows the same dependence as predicted within measurment resolution.

Spectrum taken from Nimbus ii over the equator. Note the fine structure at the centre of the CO2 absorption. There is a rise in effective temperature at the central line as predicted.

About a year ago I wrote a program to calculate the effective height at which IR photons escape to space for each wavelength in the 15 micron band while CO2 levels increase.  I indeed found a  clear logarithmic fall off in radiative forcing (DS) as  CO2 concentration increases. This can then be fitted this to a simple formula. The accepted fit  by IPCC follows that made by Mehl et al. This is

DS = 5.3\ln(\frac{C}{C_0})

where C_0 is the initial value (300ppm) and C is the new value (say 600ppm). (I actually got a bit higher than that)

Fi 1: Dependence of radiative forcing on CO2 concentration in the atmosphere. The red curve is a logarithmic fit.

Dependence of radiative forcing on CO2 concentration in the atmosphere. The red curve is a logarithmic fit.

To derive the temperature response (DT) to that forcing (DS) you need to consider Stefan Boltzmann. To cancel out the extra ‘forcing’ and rebalance energy at the top of the atmosphere, the surface will heat up so as to radiate to space exactly the same amount as before the perturbation. Crudely speaking.

DT = \frac{DS}{(4\sigma \times \epsilon \times T^3)}


DT = 1.6ln(C/C0)

Climate sensitivity (CS) is usually defined as the change in temperature after CO2 levels have doubled. So ignoring feedbacks for the moment

CS = 1.6\ln(2)

and putting the numbers in gives you then get

Equilibrium Climate Sensitivity (ECS) = 1.1C

To get the more scary scenarios as reflected in  IPCC climate models various assumptions need to be  made  for feedbacks (H2O, clouds, lapse rate, ice albedo ….). Some feedbacks like lapse rate, and maybe clouds can also be negative, but overall the net feedback is mostly from moderately to  strongly positive. Let’s call this net value of all feedbacks f. Due to the iterative nature of feedbacks (feedback acting on feedback), it turns out that the resulting temperature change DT becomes

DT = 1.6\frac{\ln(2)}{(1-f)}

f = 1 gives a runaway warming. This is a bit like guitar feedback through an amplifier where the loud speaker sound gets ampilied iteratively by the pickup.

So what is the value of f? Well if you fit the Hadcrut4 data to a logarithmic dependency plus a natural variation then you get a value

DT = 2.2\ln(\frac{C}{C0})

which gives f ~ 0.3 and the Transient Climate Response (TCR) = 1.5C. However it is quite possible that f is not actually a constant with temperature  and may even be tuned on earth to some rough equilibrium temperature. It is remarkable that the earth’s temperature has remained favorable to life for 4.5 billion years, during which time the sun’s output has increased by 30%. The obvious candidate for such a  thermostat are the oceans. They cover 70% of the earth’s surface and can both warm and cool the surface depending on temperature. The largest uncertainties in climate models are clouds and aerosols. These are the very mechanisms that could provide a natural thermostat more powerful than man’s turning up of the CO2 control knob.

You hear a lot about the debate being over and the science being settled, but that isn’t really true. There  is still a large  uncertainty about the future climate response to inceasing CO2 levels because the value of f remains unknown. We don’t even know whether it is a constant. You only have to look at the spread in climate models to see how uncertain the future is.

Graph shows 31 CMIP5 model projections up to 2100, under the highest conceivable RCP8.5 emissions scenario. Temperature ranges from 3C to 6C. The red points are Hadcrut4.4 where recent trends favour the lower range.

31 CMIP5 model projections up to 2100, under the highest conceivable RCP8.5 emissions scenario. Temperature ranges from 3C to 6C. The red points are Hadcrut4.4 where recent trends favour the lower range.

So the consensus among climate scientists is really only about the first order greenhouse effect. There are still large uncertainties about how the rest of the climate system reacts thereafter.

About Clive Best

PhD High Energy Physics Worked at CERN, Rutherford Lab, JET, JRC, OSVision
This entry was posted in AGW, Climate Change, climate science, IPCC, Physics and tagged , . Bookmark the permalink.

23 Responses to CO2 Global Warming

  1. omanuel says:

    A worldwide crash is inevitable, . . .

    if the trajectory locked in place in AUG-SEPT 1945 was aimed at:

    _ a.) A one-world, totalitarian government, in violation of

    _ b.) The public right to enjoy life, liberty and the pursuit of happiness

    See Stalin’s science

  2. Cytokinin says:

    As always Clive, I really appreciate all the hard work that you do and your excellent physical analysis. Your artical is very well written, so comprehension is considerably aided. Is it OK if I print this of for my students to read, they are unfortunately only presented with a small part of the climate story and I am trying to encourage them to develop sceptical and analytical minds

    The heat capacity of the oceans is enormous compared to the atmosphere, it must be the principal temperature regulator on the planet. I have read that water in the atmosphere accounts for sixty percent of greenhouse effect, which strikes me as amazingly precise, not sixty one percent, or fifty nine percent. It seems as if climate science has presumed that water is unimportant because the atmospheric half life of a water molecule is short, however it is surely the average amount of water in the atmosphere that effects warming and water is a much more potent gas than CO2. Any small change in water content would presumingly have much more effect than any small change in CO2. It is easy to focus on CO2 because it is considered to be a well mixed gas, but water is tricky, with huge variations across the planet, across the day and across the year. It is the primary driver of convection in the atmosphere and in the oceans, it is also how much energy is released from the atmosphere in the form of precipitation and storms.

    I read an article a couple of weeks ago about the weakening of the earth’s magnetic field. This was news to me and my immediate thought was that this could affect radiation penetrating into the atmosphere. I was able to quickly find research which shows that this has been happening and that there has indeed been deeper penetration of cosmic rays. This has been happening since 1840 and has been accelerating over the last two or three decades.

    Thinking about post industrial changes, several things came to my mind that could affect climate. The growth of urbanisation and the changes to the world resulting from this and population growth. As well as pumping out gases into the atmosphere, industry has pumped out water vapour. It has also pumped out massive quantities of warm water into the oceans. These are of course small increments of energy in comparison to the effects of solar inputs, but small increments over a long time can have considerable effect. Warmer water evaporates more readily and so can lead to more warning or cooling depending on circumstances.

    Irrigation must allow increased evaporation, and I see that there are indeed studies to show this to be the case, again the effect can be warning or cooling. Of course any local effect will have knock on effects in other parts of the atmosphere, since all systems move towards equilibrium.

    Cities are considerably warmer than surrounding countryside, they also generate local greenhouse effects that means they remain warmer during the normal night cooling processes. This is more energy into the planetary system.

    There are now millions of miles of tarmac roads. The energy absorbing and emitting qualities of roads are quite different from that of surrounding countryside. Again this must have influence on heat distribution. I have felt how roads can be warmer than surrounding land long after sundown. This must have a dampening effect on the normal diurnal cooling cycle.

    The burning of fossils has pushed CO2 into the atmosphere. This extra carbon has mass and so must have increased atmospheric pressure slightly. We still measure temperature at ground level, but should measure at a slightly higher level to compensate for this increased pressure. One atmosphere is no longer one atmosphere, it is one atmosphere and a wee bit. Again it is an increment, but possibly important.

    The last century and a half has seen increasing amounts of groundwater being utilised for irrigation, drinking water and industry. This is water that otherwise would not have entered the hydrosphere and which could be influencing climate.

    Post industrialisation, how much energy has been pumped into the oceans and atmosphere? How does this compare to back radiation from CO2? Has this energy remained in the oceans?

    I am interested in the full picture because we seem to have become fixated with CO2 as the primary driver of climate change. If the primary driver is something else then we need to identify what, since cutting CO2 emissions could have zero or minimal effect. If the primary drive is a natural system e.g. increased solar or cosmic radiation, then there is nothing we can do except watch and learn. If it is the result of human activity is it possible to change behaviour? How do you stop industry, cities, roads, irrigation, groundwater usage? Seven billion people want to live like Americans, not Amazonian Indians.

    People continually tell me that we need to reduce CO2 emissions now. When I say what if it’s not CO2 then they argue that it can’t do any harm to cut emissions and cite the precautionary principle. What if you are sticking a finger in the dyke and the sea is flowing in round the edge?

    • Cytokinin, obviously your knowledge on water vapor is less than zero / it’s back to front, your students deserve the truth: somebody should advise them; to compare the desert where is no WV and jungle, forests, where is rice paddies = lots of water vapor = GOOD CLIMATE, cooler days warmer nights!!: They should see for themselves that: in deserts on same latitude is 5-10C warmer days / colder nights – is called BAD CLIMATE.

      Your students should read this post, not adulterated by the Warmist propaganda::

    • Clive Best says:


      Feel free to use it as you wish. I originally wrote this for a friend who asked me to help with an article he was writing about the diminishing warming effect of CO2 with concentration. Then I decided to post it here.

      The earth is a water planet with 70% of its surface covered by oceans. That is the main reason we have such a stable climate, and why there are no extremes of temperaure at the equator. The hottest places on earth are the dryest regions where Hadley cell dry air descends. A band of clouds cover the tropics and follow the sun with the seasons. This shades the surface and cools the oceans through evaporation and convection. Water temperatures never exceed 30C. The fact that the earth has had liquid oceans for over 4 billion years during which time the sun’s output has increased by 30% proves that they must act as a thermostat.

      The earth’s magnetic field actually reverses at regular intervals every few million years or so, and we are about due for another reversal. I was unaware that it was currently decreasing.

      The impact of industrialisation, development and a population explosion on the planet has been huge. CO2 is just one aspect. The Urban Heat effect is real and measurable on the land surface. Strangely enough it appears to cool the past rather than warm the present. The reason for that is that each weather station is normalised to a period 1961-1990 and we just measure ‘anomalies’ relative to that. Major cities developed fastest before 1990 so this warming is subtracted off the future and therfore makes temperatures before 1990 look cooler than the surroundings. This effect is not well known.

      We cannot stop everyone on earth from burning fossil fuels. We can pass laws that force western countries to reduce emissions but that will have little effect on the climate unless Asia follows suit and the devloping world stops developing. Unfortuanetly easily extracted fuels will all eventually be burned no matter what our governments do. Every year Indonesians burn down the rain forest to plant Palm oil causing a smog over the area. The palm oil is used to make Biodiesel which they sell to us, so we can feel smug and green. London buses proudly boast about being biofueled.

      I am pretty sure that climate sensitivity is low – about 1.5C and certainly less than 2.5C so global warming won’t be a disaster anyway. It may even be beneficial, especially if it delays the next Ice Age by a thousand years.

      Solar energy makes sense for isolated communities and semi-independence from the grid. However long term we have to find an alternative high density power source if we want to maintain modern life. The only solution which makes sense is nuclear power. Ideally Nuclear Fusion can be made to work, which is safe and free of long term radioactive waste. Furthermore there are no nuclear proliferation worries so it can be deployed anywhere.

      • Ron Graf says:

        Clive, your post could be the textbook on this. I am marking it as a reference to refer people to for separating the “settled science” part from the under investigation part.

        You say:

        Major cities developed fastest before 1990 so this warming is subtracted off the future and therefore makes temperatures before 1990 look cooler than the surroundings. This effect is not well known.

        How do you suppose this would affect the splicing of proxy temperature reconstructions onto the instrumental record? Could this account for the divergence problem when the proxies enter the 20th century? For example the recent Ocean2K study had multiple proxies fail to rise significantly when entering the 20th century although they have been cross-calibrated and deemed robust proxies. Here is the recent CA post on Ocean2K.

        You also bring up a good point about whether CS has a temperature curve sloping down. I had asked this question of Nic Lewis at CA a few months ago. He said he thought not but I realized in his next post he had a bone to pick with CS variability since some IPCC models he showed increase sensitivity with temp, they curved up.

        • Clive Best says:


          Thanks. I think there is a renormalisation problem with using temperature anomalies, which is simply papered over. Anomalies are calculated relative to some arbitrary ‘normal’ period. HADCRUT use 1961-1990, GISS use 1959-1980 and NCDC use the full 20th century. As far as Ocean2K goes I cannot find out how they define their temperature anomaly.

          It is common practice to simply add or subtract an offset so as to align anomalies from different sources. The temptation is to slice proxy data onto the back of the measurement data so as to tell a simple story. In reality there are large errors on any such renormalisation which are probably as large as 20th century warming itself.

          My estimate is that the urban heat effect is responsible for about 0.2C of the 1.0C warming in CRUTEM4.

      • Fernando L. says:

        The “easily extracted fossil fuels” is a huge looming problem. The way the market works, the marginal, very high cost sources set the price. USA “shale”, extra heavy, deep water, high water cut, North Sea, and old stripper wells are the marginal tranches, and they need ever increasing prices. Eventually prices have to increase so much the renewables or nuclear power will have to kick in. And the prices will probably be going roughly triple today’s levels in 15 years. This means we just can’t reach those high production levels seen in RCP8.5.

        I keep hammering at this point because we see a lot of work being done on a bogus figure. That 8.5 was invented by the IPCC. It wasn’t really estimated using a reasonable set of asumptions.

    • A C Osborn says:

      “When I say what if it’s not CO2 then they argue that it can’t do any harm to cut emissions”

      They are wrong of course. The Earth needs more CO2, not less, it is plant and plankton food.

      • Clive Best says:

        Yes it does harm us.

        Encouraging motorists through tax breaks to buy diesel cars was supposedly to cut CO2 emissions. The end result was that thousands of people have died as a result in major cities because of ideology. CO2 is a harmless gas whereas NO2 is toxic. The CO2 emission savings were anyway negligible.

      • Osborne – IF CO2 didn’t intercept and absorb sunlight – water, clay, or rocks would have done that same job – sunlight doesn’t go trough the earth, to come from the other side, if it wasn’t co2.

        2] water ”absorbs” sunlight / heat – during the day and releases it during the night, that’s how it makes BETTER CLIMATE! ”Absorbing heat” is considered bad by the Flat Earthers, who think that is sunlight 24h everywhere…!!! CO2 is the good guy – carbon molesters are naughty…

  3. Ron Graf says:

    To round out the other aspect of CO2’s warming power, it residence time in the atmosphere is dependent on oceans. I just noticed a chart recently that showed CO2 still on the rise for the last 2000 years up to the 19th century, up from about 270ppm to about 280ppm. This means the Holocene started with under 200ppm (when it was warmer). And, it would indicate that, although temperature had been gradually dropping since the Holocene optimum, 8k yrs ago, and oceans were still oversaturated and emitting CO2 before the 19th century. If this is true then the saturation point at GMST 0.8C lower than now is higher than 270ppm.

    The IPCC says that 28% of mankind’s added CO2 is currently being dissolved into the oceans. So the saturation point at current GMST I would guesstimate as 300ppm. I have not seen the IPCC position of that the steady-state concentration curve is for atmospheric emissions versus ocean uptake but I have read blogger calculate that at current emissions CO2 will peak at between 500ppm and 550ppm in 30-60 years. So with 560ppm being double the 280ppm of pre-industrial atmosphere we should see 1.5C warming as the highest net result if Clive’s ECS is correct and 1.5 to 4.5C if the WWF and Greenpeace people are correct. (This is assumes that we do not go over 560ppm which assumes that we will not completely ignore alternative technologies and find endless cheap fossil fuel.)

    The other good news is that once we find alternative energy the atmosphere CO2 will continue to decline on a fading curve as it sinks to the ocean, which will act a reservoir to stabilize at a beneficial amount of CO2 in the air. Remember, without CO2 we would certainly resume our slow temperature decent toward the tipping point of ice-albedo feedback induced ice-age for which we would not be scheduled to emerge for 100K years. We forget the Holocene is but an interglacial pause (mostly done now) in the Quaternary Ice Age (current).

    • Fernando L. says:

      I estimated 630 ppm. But that depends on renewables and nuclear being able to fill a growing gap. If they can’t all bets are off. It’s a very difficult system dynamics problem. I don’t think it can really be simulated.

  4. omanuel says:

    We are nearing full disclosure to seventy years of deceit (1945-2015).

    In fact, Sir Fred Hoyle “blew the cover” on Stalin’s biggest lie in his 1994 autobiography:

  5. In 2004 the Mauna Loa CO2 was 377.6 and in 2014 it was 398.5. Plugging those values into your no-feedback formula you get a temperature rise of 0.09C for the decade. Using your feedback formula with f=0.3 this increases to 0.12C which is the sort of temperature rise that the models, or their ensemble mean or somesuch, predict. This rise has not occured. But do we just assume that “natural variation” masks a rise which will return soon? Is there any way in the short term of testing these various predictions, or do we just have to wait and see? The apparent lack of testability of these climate predictions is amazing considering the importance that is attached to them.

  6. Ryan says:

    The reality is the climate’s around the world are not static and never were. Just look at the last 40 years. It started with the ice age scare in the 70’s and in 10 years turned to global warming and 15 years later when the warming stopped it is now climate change. The scam going is what ever direction the climate goes, they will find some way to blame it on human activity. It is obvious now that CO2 is not a driver for temperature. They are finding the rising C02 levels are the result of the tiny warming that has occurred and lags behind temperature rise. If these people are right and the temperature remains the same, the rise of C02 levels should slow and eventually plateau. Also reality is we are living the positive affects of higher C02 as it increases vegetation/food growth. Obviously the IPCC will not want to shed light on positive C02 affects. They need to keep global funding coming in to be able to afford to continue their world wide fraud. The world climates are too complicated for anyone to fully understand and need a lot more research. but they are using gullible human superstition and fear of the unknown making belief in man affected climate change no different than bowing and worshiping a piece of wood carved into the likeness of a god.

  7. Victor G. says:

    The nature of CO2 absorption at low pressure seems more complicated. Without collision-mediated energy dissipation, the ability of CO2 to absorb IR photons will be greatly diminished due to saturation. This effect would be further exacerbated by narrowing of the CO2 absorption lines, which would make them even more prone to saturation. As a result, the effective CO2 absorption will start deviating (to lower values) in low-pressure layers of the atmosphere from what is expected based purely on its concentration change with altitude. So the troposphere height increase may be somewhat lower that what is predicted by the above model, and therefore the CO2 impact may be smaller.

    This effect seems to be playing out on Mars, where 10x larger amount of CO2 is unable to create any significant greenhouse effect.

    • Clive Best says:

      At very low pressures, the mean free path of a CO2 IR photon is large enough to escape to space so the lapse rate stops. To get a greenhouse effect you need a lapse rate generated by radiative transfer. So the height of the troposphere depends on CO2 and H2O concentrations.

      • Victor G. says:

        Yes, of course. I am just saying that at very low pressure, the effective attenuation of CO2 maybe somewhat lower than what we would expect based on purely the concentration change with altitude. That would make the upper layers of the troposphere more transparent to IR compared to what these simple models predict, affecting the troposphere height increase and potentially reducing the GHE.

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