Forcing and Burning in Coal Country.

Guest Post by Willis Eschenbach

A 2015 paper by Xiaochun Zhang and Ken Caldeira has come to my attention. Inter alia, the abstract says:

For example, the global and time‐integrated radiative forcing from burning a fossil fuel exceeds the heat released upon combustion within 2 months. Over the long lifetime of CO2 in the atmosphere, the cumulative CO2‐radiative forcing exceeds the amount of energy released upon combustion by a factor >100,000. 

That made my bad number detector start flashing red. So I decided to do my own rough calculations. Here’s my work. I’ve used values for the amount of carbon which would be needed to be burned in order to raise the atmospheric CO2 by one part per million by volume (1 ppmv)

BURNING CALCULATIONS 

2.1E+12 — kilograms of airborne carbon (as CO2) per ppmv of CO2

43% — airborne fraction, the amount of the CO2 emissions that remain airborne

5.0E+12 — kilograms of emissions of carbon needed to raise atmospheric CO2 levels by 1 ppmv

70% — average carbon content of coal

7.1E+12 — kilograms of coal burned per ppmv of CO2 increase

2.4E+07 — joules per kilogram, energy content of coal

1.7E+20 — total joules per year from burning 7.1E+12 kg of coal

FORCING CALCULATIONS 

0.013 — additional forcing in watts per square meter (W/m2) when CO2 goes from 400 to 401 ppmv

420,608  — convert watts per square meter to joules per year per square meter

5.1E+14 — square meters, surface area of earth

2.1E+20 — total joules per year from 1 ppmv additional forcing

So in year one, CO2 radiativeforcing gives about 30% more energy than we got from burning coal. 

And in the following thousand years, depending on the carbon model chosen (IPCC Bern model, or Joos model as in their paper), we end up with between 250 to 400 times the energy from the CO2 radiative forcing as from the burning of the coal.

Now, recall that the claim was that the “cumulative CO2‐radiative forcing exceeds the amount of energy released upon combustion by a factor >100,000”.

And I got a factor of 250-400. So my question is … have I made an error, and if so, where? Wouldn’t be the first time …

Notes:

Global coal consumption is about 8E+12 kilograms per year. Coincidentally, this is also about the amount of coal shown above as being needed to increase airborne CO2 by 1 ppmv.

I do not think that an increase in CO2 forcing perforce means a temperature increase. I think it is counterbalanced by changes in emergent phenomena that counteract the slight change in radiative forcing. 

I call it a “slight” change in radiative forcing because I divide the phenomena that affect some given system into 1st, 2nd, and 3rd order variables.

What I call “first order variables” represent and can change more than ten percent of a signal. You generally need to include these in even an initial analysis. They are large enough to be significant.

Second order variables make up from one to ten percent of a signal. You need to include these variables in any more detailed analysis of a situation. 

Third order variables represent less than one percent of the signal. They are lost in the noise, and can be neglected in any but the most exhaustive and detailed analysis.

And how does this question of variable types apply to the annual global coal burning issue?

Global average downwelling radiation (solar plus longwave) is about five hundred watts. A year of global coal burning gives a hundredth of a watt per square metre change in this half-kilowatt system. 

That’s about two thousandths of one percent of the signal. Third order.

How about a longer-term effect? Well, it’s possible that by the year 2100 we’ll see CO2 levels double from the present. Or not. Here are some representative supply-driven scenarios:

ORIGINAL CAPTION: Comparison of atmospheric CO2 concentration under SD [supply-driven] scenarios with those from a range of current literature that examines ‘supply-driven’ fossil fuel emission scenarios. (SOURCE)

Note that none make it But heck, for the sake of discussion let’s assume that technology doesn’t progress and nuclear is ignored and at some point in the next eighty years the CO2 level doubles. That would increase downwelling radiative forcing by 3.7 W/m2 … which is still only three-quarters of one percent of the total signal. Third order.

Here’s the difference. Clouds are a first order variable regarding the global energy balance. They can move the downwelling energy up and down by hundreds of watts in minutes. That’s on the order of fifty percent of the total five hundred watt average signal. Over ten percent, first order.

CO2 changes, at the other end of the spectrum, are a third order variable. Even an improbable doubling represents a change of less than one percent of the five hundred watt system. Lost in the noise. Counteracted by a small change in cloud emergence time and prevalence.

But I digress … so let me ask again:

Where is the error in my calculation of thermal versus radiative forcing of coal? I get a very different answer from that of Zhang and Caldeira.

Best to all. I’m still doing building construction in Alaska, near the Kenai River. A moose wandered by the window the other day. Yesterday afternoon it was a bald eagle parting a gaggle of seagulls. What a place! 

And of course, it being solstice, it’s never too dark to read headlines in the newspaper.

Warmest midsummer (or midwinter) wishes to everyone, podal and antipodal,

w.

As Usual: When commenting or pointing out my error, please quote the exact words that you think are wrong. Only in that way can we be clear about your meaning.

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72 thoughts on “Forcing and Burning in Coal Country.

  1. 2+ orders of magnitude error…. sounds about par for the course for consensus climate science.

  2. “the global and time‐integrated radiative forcing from burning a fossil fuel exceeds the heat released upon combustion within 2 months. Over the long lifetime of CO2 in the atmosphere, the cumulative CO2‐radiative forcing exceeds the amount of energy released upon combustion by a factor >100,000.”

    Again, that’s a completely meaningless statistic, even if true. The actual number you’re looking at is insignificant. The study at issue calculated that burning one exajoule of coal would contribute a mere 0.000124 watts per square meter. A single exjoule is the equivalent energy contained in 163 million barrels of oil. Who cares whether that 0.000124 W/m2 lasts so many centuries that the sum of the energy from the greenhouse effect over such a long period of time is 100,000x the energy you got from burning it? We still got a huge benefit from burning that 163 million barrels of oil, while the 0.000124W/m2 that results from that specific expenditure does no harm to anyone, no matter how long it lasts.

  3. RE: “Global average downwelling radiation (solar plus longwave) is about five hundred watts. A year of global coal burning gives a hundredth of a watt per square metre change in this half-kilowatt system. That’s about two thousandths of one percent of the signal. Third order .”

    Boom! There it is! That is the most damning and irrefutable evidence against the global Catastrophic Climate Change fraud.

    • You’re exactly right. And the total contribution of all CO2 added since the industrial age is asserted by the IPCC to be less than 2W/m2, or less than one half of one percent of the 500W/m2 already hitting the surface of the Earth. Engineering students are taught that anything beyond the third significant digit can be discarded as irrelevant, which would mean that anything less than 1% variation can be ignored when determining the behavior of a system.

  4. If that exajoule of energy is released over a period of 1 year, then the amount of additional energy retained in the system is 100X that (according to Caldeira et al.) per year. We have found a way to increase the efficiency of burning by two orders of magnitude! Now all we have to do is figure out a way of extracting that ‘free’ energy. How about thermocouples to generate electricity? It looks like it might be profitable to burn coal just for the CO2 and not worry about any short-term applications such as boiling water. /sarc

  5. It doesn’t pass the smell test. With all the vegetation fires over the surface of the Earth in the past 10,000 years, that must amount to several times the total surface, plus all the GHGs spewed by volcanoes the accumulated warming would have cooked the entire surface of the Earth.

    So either the CO2 doesn’t last in the atmosphere for thousands of years, or doesn’t produce the calculated warming, or neither of them.

    • I think the only way CAGW works is if CO2 has a very long residence time in the atmosphere. The thing is that we’re seeing a greening planet. NASA That means more CO2 uptake by the environment. So, long CO2 residence times are rubbish.

      • The US and the USSR did a very interesting experiment in the 1950s. The shot off a large number of nuclear weapons in open air, thus seeding the atmosphere with a charge of C14. If the changers are correct, most of that C14 should still be in the atmosphere as it has half life of ~5730 years.

        Is it? Has anybody tried to measure it?

        • Wikipedia has a graph for New Zealand and Austria. Sorry, I don’t know how to cut and paste graphics. It shows a spike then quite a fast falling away (as you would hope and expect). CO2 may be the same since it and its chemical derivatives are quickly deposited on and in the benthos in the form of dead marine organisms. There is a great deal of sequestered CO2 in the earth in the form of limestone rock. We may be in a limestone forming period which would mean relatively quick clearance of excess CO2 from the atmosphere.

        • If the changers are correct, most of that C14 should still be in the atmosphere as it has half life of ~5730 years.

          Carbon turnover rate is completely independent of concentration growth.

          For example If 1000ppm is removed from the atmosphere and 1100ppm is added to the atmosphere, the net gain is 100ppm but the C14 concentration is lower.

        • Adding 1 ppm CO2 to the atmosphere will “accumulate” in the same way that adding 1 ppm water to a river “accumulates” in the river.
          Or using fluxes, for example, adding 1 liter per second to the flow of a river of 99 liters per second will result in a flow rate of 100 liters per second. The newly added water never accumulates. A river always has an infinite sink. The new water is always 1 percent of the total flow of the river.
          Basic biogeochemical carbon cycle.

          The 14CO2 spike added by atmospheric nuclear testing fell to 1/2 in ten years.
          The peak was 1964 in the northern hemisphere, and 1965 in the southern hemisphere.
          50 percent of the 14CO2 was gone by 1975. 3/4 gone by 1985, etc. for a Tau of 20 years, or 5 percent per year.

          https://wattsupwiththat.com/2013/07/01/the-bombtest-curve-and-its-implications-for-atmospheric-carbon-dioxide-residency-time/#comment-1352246

          Adding “new” CO2 at the rate of 4 percent to the flow of “old” CO2 through the atmosphere results in a flux that is 4 percent higher than before. The newly added CO2 flows to the sinks just as the old CO2 flows into the same sinks. The atmosphere will remain 4 percent “new” CO2, since the sinks never discriminate between “new” and “old” CO2

        • Walter Sobchak,

          “How long does it take for nuclear fallout to disappear?

          After 7 hours, the residual radioactivity declines 90%, to one-tenth its level of 1 hour. After 49 hours the level drops again by 90% and after 2 weeks it drops a further 90%. By the time 14 weeks has gone by the radiation will be 1/10,000 the level as measured 1 hour after the detonation.”

          https://www.google.com/search?q=US+in+the+1950s.+shot+off+a+large+number+of+nuclear+weapons+in+open+air%2C+thus+seeding+the+atmosphere+with+a+charge+of+C14%2C+most+of+that+C14+still+in+the+atmosphere&oq=US+in+the+1950s.+shot+off+a+large+number+of+nuclear+weapons+in+open+air%2C+thus+seeding+the+atmosphere+with+a+charge+of+C14%2C+most+of+that+C14+still+in+the+atmosphere&aqs=chrome.

          • Johann Wundersamer

            Those values are for “nuclear isotopes (daughter products) remaining in the atmosphere after a nuclear blast”, not for “C14 remaining in the atmosphere after a nuclear blast”.

            Half-life for C14 is “5,730 years. The half-life of a radioactive isotope describes the amount of time that it takes half of the isotope in a sample to decay. In the case of radiocarbon dating, the half-life of carbon 14 is 5,730 years.”
            As above, from Wikipedia. Far more C14 is removed from the atmosphere by natural absorption into trees and plants after an airborne/surface blast than by half-life decay.

      • But the processes that green the earth take CO2 out of the air and do reduce residence time.
        There are then two extreme cases to consider. At one extreme, there is only a tiny weight of global vegetation to be greener. It can go as green as possible in a very short time, but it uses up only a tiny part of the total CO2 in the air.
        At the other extreme, there is so great a mass of vegetation to be greened that the amount of CO2 in the air is insufficient to make everything the greenest green. At this stage we get into the kinetics, like the rate at which vegetation can extract CO2 from the air in different seasons and annually overall.
        Then we are back to modelling being the best way to attack the problem, raising again the old debates about bomb test rates, source and sink mechanisms and kinetics, etc. There remains so much yet to be measured with sources and sinks that the modelling results have large uncertainty brackets for the time being – so large that endless arguments can rattle on within the uncertainty brackets. So much for settled science. Geoff

    • It matters not one (1) twit how long a CO2 molecules remains in the atmosphere after being released therein, be it for ½ second or 1,000 years. Raindrops are quick at stripping CO2 molecule from the atmosphere, and so is surface waters and plant stomata.

      And any thermal “heat” energy (LWIR) that a CO2 molecule per chance might absorb, is not cumulative from one minute to the next, one hour to the next and surely not one year to the next.

      And there is no Natural Law that states ….. “all LWIR that is radiated from the earth’s surface must be absorbed by a “radiant gas” molecule and partially re-radiated back toward the surface ….. before it can finally be radiated back into outer space”.

      Prove the above hypothesis wrong. Take a handheld LWIR detector out to the Arizona desert on an extremely hot and dry afternoon ….. and measure the amount of upswelling (radiating) LWIR @ 24 inches above the surface, ……. and then turn your LWIR detector over and measure the amount of downswelling (re-radiated) LWIR @ 24 inches above the surface,

  6. These alarmists appear oblivious to the way the atmosphere actually works. Well above the tropopause, radiation to space is primarily from CO2. If you assume there has been no increase in water vapor (big mistake), WV averages about 10,000 ppmv. The increase in absorbers at ground level since 1900 is then about 10,400/10,300 = ~ 1%. The increase in emitters to space at high altitude (~> 30 km) accounting for the lower atmospheric pressure is 432/332 * 0.012 = ~ 1.5%. This easily explains why CO2 increase does not cause significant warming and might even cause cooling.

    • “The increase in emitters to space at high altitude (~> 30 km) accounting for the lower atmospheric pressure is 432/332 * 0.012 = ~ 1.5%”

      Can you explain this please?

      • The low temperature (~ -50 °C) at the tropopause limits the WV to about 32 ppmv. CO2 is about the same ppmv at all altitudes (~ 300 ppmv in 1900 and ~ 400 ppmv now). Emitters = sum of WV and CO2. Pressure at 30 km is 0.012 atm. The increase in absorbers at ground level is countered by increase in emitters at high altitude.

    • This is not true. The entire atmosphere radiates to space. CO2 traps and re-emits 15 micron radiation, which is originally emitted by the atmosphere where it is around -80 C. Up there where the atmosphere is cold, CO2 prevents radiation to space below SOME altitude. Increased CO2 raises this altitude.

      Pretending that this can be calculated in the first place, and describing it as a known flux about which we can debate, is ludicrous. Don’t drink the Kool-Ade. The atmosphere up high cannot heat the surface of the Earth, nor heat itself. Think about the CO2 at TOA as preventing heat escaping to space. More CO2, higher altitude at which the atmosphere freely radiates to space, lower temperature at which the atmosphere freely radiates to space, less energy radiated to space.

      There is a lot of bad physics bandied about and taken seriously by “Climate Scientists,” a LOT.

  7. Willis,
    The simple answer would appear to be that you are using too short a lifetime for CO2 in the atmosphere.
    Looking at the supplementary information and in particular Eq. S1 and Fig. S1 shows that the authors
    use a sum of exponential decays for CO2 with the longest timescale being 100 000 years for weathering
    of rocks. The vastly different timescales between your calculations and theirs easily explains the different numbers.

      • No it doesn’t Willis simply plucks a number for the lifetime out of the air with no justification. If he wants to argue about why their calculation is wrong then he needs to discuss their model rather than just ask where the error is.

    • Izaak, thanks for that. I didn’t pick 1000 years at random. I took it because their graphs stopped at 1000 years.

      However, there are a couple of problems with the idea that the 100,000-fold number is meaningful. First one is that if I add it up out to 100,000 years from now, I still only get ~ 28,000 times the burning energy for the value of the CO2 radiative forcing, not 100,000 times the energy.

      The deeper problem is the carbon model used. They use the Joos carbon model for the decay of a pulse of CO2. Here’s the problem with the Joos model. Below is the prediction of the Joos model of the decay of a pulse of CO2 over 100,000 years.

      I’m sure you can see the difficulty. The Joos model says that when a pulse of CO2 is emitted into the atmosphere, about twenty percent of that pulse remains in the atmosphere forever. The system never returns to the status quo ante.

      I find that extremely implausible. Consider. If twenty percent of every pulse of CO2 into the atmosphere from every volcano since time immemorial was still in the atmosphere, wouldn’t the CO2 level be through the roof? It’s a permanent escalator cause, always increasing the CO2 concentration. I do not see how that is physically possible

      I say that if you emit a single pulse of CO2 into the atmosphere with no further emissions, the system will settle back to where it was before. Not to a higher level. To the previous level.

      I’ll also say that when the authors claim that some increased forcing lasts forever, then any comparisons are totally meaningless. The more years they pick to add up, the greater the disparity and the bigger the scary numbers. The less years they pick, the less the disparity.

      I will note that the same problem exists with the carbon model mostly used by IPCC authors, the Bern carbon model. Like the Joos model, it says that the CO2 levels will stay elevated forever after a single pulse of CO2. It just says that a smaller amount, only fourteen percent of the CO2 increase, persists indefinitely.

      But that suffers from the same logical flaws whether it’s fourteen or twenty percent.

      There is a further difficulty. They make the claim that in two months the radiation from the CO2 has equalled the energy from the burning of the coal. This means in a year the increased CO2 radiative forcing would be six times that of the burning. I only get 1.3 times the burning. They’re four times as large. I can’t figure out why.

      Anyhow, Izaak, that’s how it looks upon a second look. Yes, if you add up a hundred thousand times twenty percent of some value, you’ll get a big number. So what? I say that has no physical meaning.

      My thanks to you, sir, for pointing out that they are looking out 100 kiloyears and I was just looking out a “mere” thousand years into an unknowable future …

      w.

      • Step 1. Replicate what they did.
        Step 2. Do your own version and explain what is wrong with theirs and why yours is
        better.

        Lastly

        “I say that if you emit a single pulse of CO2 into the atmosphere with no further emissions, the system will settle back to where it was before. Not to a higher level. To the previous level.”

        needs more argument than merely “I say”

        • Mosh,
          In my view, the argument was settled in the paragraph above the “I say”. It is the “infinite escalator” argument. If every CO2 pulse for the last several billion years left a percentage of its pulse in the atmosphere, that percentage would have to be vanishingly small for the CO2 to be measured in parts per million today. The presence of life here is evidence of that. If O2 is too low, animals disappear. If CO2 is too low plants disappear. The fact that we animals have successfully coexisted with plants as long as we have says, to me, that the”infinite escalator” argument is adequate. The residual may be there, but it’s vanishingly small.

        • Steven Mosher June 22, 2019 at 2:43 am Edit

          Step 1. Replicate what they did.
          Step 2. Do your own version and explain what is wrong with theirs and why yours is
          better.

          Thanks, Steve. Hard to do, because I couldn’t find out how many years into the future they ran their bogus lunacy.

          Lastly

          “I say that if you emit a single pulse of CO2 into the atmosphere with no further emissions, the system will settle back to where it was before. Not to a higher level. To the previous level.”

          needs more argument than merely “I say”

          Sure. Le Chatelier’s principle. Experience of other disturbed steadystate systems. However, I would argue that the shoe is on the other foot. They’re making a claim that there exists an infinte escalator. They say every time a volcano goes off, 20% of the CO2 emitted never leaves the atmosphere.

          Until they can explain how that would not lead to millions of years of increasing CO2 levels, it is them who are basing their argument on “I say” …

          Best regards,

          m

          • Willis,
            Le Chatelier’s principle doesn’t imply that. All it states is that a system will adjust to
            dimish the change not that it is completely counteract it. Suppose for example I have a
            sealed container that is half filled with water and half with air. In equilibrium a fraction R
            of the CO2 will be dis-solved in the water and (1-R) will be in the atmosphere. If I suddenly increase the amount of CO2 in the container then the amount dissolved will increase but there will still be more CO2 in the air than previously even if I wait for thousands of years.

          • “Thanks, Steve. Hard to do, because I couldn’t find out how many years into the future they ran their bogus lunacy.”

            write to ken. or have zeke ask him.

            “Sure. Le Chatelier’s principle. Experience of other disturbed steadystate systems.”

            err that’s not an argument, you’d have to explain why that prinicple applied in this case. As for your experience with other disturbed systems same thing.

            these are just restating the “I say so” argument in other terms.

            and I suppose they have the right to do the same thing.

            that is, in the end the final numbers depend upon a couple of factors.
            time and the final amount that stays in the atmosphere.

            Perfect time for a parameteric study especially when you have contentious assumptions
            that are hard to settle ( I say versus he says )

            psst see Izaak

        • “…Step 2. Do your own version and explain what is wrong with theirs and why yours is
          better…”

          “…If he wants to argue about why their calculation is wrong then he needs to discuss their model rather than just ask where the error is…”

          I don’t know how accurate “…4.5 × 10^10 J of global warming per mol CO2 released to the atmosphere…” is. But let’s pretend that it is acceptable for the sake of argument.

          The authors divide that by “…393.51 kJ/mol (standard enthalpies of formation)…” to get ~100,000.

          As I posted elsewhere, this seems to be poppycock. This is simply C + O2 -> CO2 and incomplete for the combustion of coal.

          • C + O2 -> CO2 + 8084 Kcal/ Kg of carbon (33940 KJ/Kg)
          • S + O2 -> SO2 + 2224 Kcal/Kg of sulfur (9141 KJ/Kg)
          • 2 H2 + O2 -> 2 H2O + 28922 Kcal/Kg of hydrogen (142670 KJ/Kg)

          (SO2 and H2O have their own influences and residency time in the atmosphere to account for if we want to go the full monty).

          Using a molar basis of carbon as the comparison and the enthalpy of formation of CO2, burning coal releases as much energy on a molar basis as burning glucose, plant material, or anything else that involves C + O2 -> CO2.

        • Mosher: “needs more argument than merely ‘I say that if you emit a single pulse of CO2 into the atmosphere with no further emissions, the system will settle back to where it was before. Not to a higher level. To the previous level.'”
          Mosher, Willis has already provided a firm argument supporting his questioning of the supposition that 20%, or 14%, of the CO2 currently being added to the atmosphere will be a permanent addition. He points out that volcanoes have been contributing to the atmosphere since time immemorial and suggests that the CO2 content of the atmosphere would have gone through the roof by now if 14% to 20% were permanently retained in the atmosphere. Perhaps you are suggesting that the amount of CO2 contributed by volcanic activity over the earth’s 4 x 10*9 years history is not significant?
          It may be helpful to consider the earth’s atmosphere, hydrosphere and biosphere as a single system, within which exchange occurs between the component parts. Much volcanic activity is submarine, and gases emitted by submarine volcanoes go initially into aqueous solution.
          Currently about 40 times as much CO2 is present in the hydrosphere as in the atmosphere.
          How much of this came from volcanoes? Virtually all of it. How do we know? One way is to consider the amount of CO2 currently locked up in sedimentary rocks (limestones, dolostones, coals, hydrocarbons etc). This was estimated by Rubey (1951) as about 92 x 10*21 grams. All of this CO2 was emitted by volcanoes and passed through the combined hydrosphere-atmosphere-biosphere system before being sequestered as rock. For comparison, how much CO2 is currently held in the combined system? According to Rubey, about 0.15 x 10*21 grams. That is about 0.16% of the total emitted – 2 orders of magnitude less than the fraction remaining according to the Joos model. I think Willis has made his point well.
          Rubey, W.W. 1951 Geologic history of sea water. Bull. Geol. Soc. Amer. 68, 1111-1147.

      • Willis,
        They do not use the Joos model like the one you suggest. If you look at Eq. S1 it is a weighted
        sum of exponential decays with no constant term. Hence it does not suffer from the problem you mention and 0% of a pulse of CO2 will remain in the atmosphere. Again looking at Fig. S1 you
        can see how a pulse of CO2 decays to zero over time.

        So if you want to claim that the author’s calculations are incorrect you need to start with the
        same model that they use and try and replicate their results. If on the other hand you want to
        claim that their conclusion are false you need to say why their model is wrong. You appear to be
        using a different model and then saying that the author’s calculations are wrong because they got
        a different result to you something which is not logically valid.

        • Thanks, Izaak. They say they used it, and they gave the exact formula I used.

          Later on they also say that they used a model without a constant term … which is a pathetic attempt to replicate the constant term without having one. Instead, they use an equation that, no exaggeration, has a decay period of 352,858 years … which of course is basically constant in any but the longest time span.

          C’mon, pull the other leg. Are they sure the period is not 352,857 years? They say:

          Time constants t1 = 1.9 year, t2 = 90.1 year, t3= 269 year, t4 = 10,739 year, and t5 = 356,878 year, correspond roughly to time scales for carbon to mix into the oceanic mixed layer, thermocline, and interior, and time scales for neutralization by carbonate and silicate mineral weathering, respectively.

          That’s crazy. I’ve never figured out why they would think that 10% or so of the CO2 in the atmosphere would wait around for hundreds of thousands of years just to weather silicate minerals. I see no reason that any of those processes should apply to a fixed percentage of the atmospheric CO2. Why would CO2 molecules not be sequestered by a shorter term sink long before that 356,878 years had elapsed?

          w.

          • Willis,
            Just because you think something is “crazy” doesn’t make it wrong. I have no idea about the validity of their model but it would assume that each sink has a finite capacity and reaches equilibrium at different timescales. It is also well known that over geological timescales CO2 is regulated by the weathering of rocks so that explains the very long timescale of that process.

  8. Willis, You say the factor is 250-400. That’s roughly 1E2.5. Zhang and Caldeira say it’s 1E5. Well, in Global Warming math, 5 is double 2.5, so they’re only off by a factor of 2. That’s spot on, in their world.

    • Tom: Much larger than 2. The engineering notation 1E2.5 is 1×10^2.5=316; 1E5 is 1×10^5= 100,000. The latter large figure is = 316^2, or 316 times the smaller number. Twice 1E2.5 would be 2E2.5.

  9. About 3 years ago now a couple of papers, one Japanese, the other Chinese I think, found that any increase of CO2 in the atmosphere caused a similar reduction in water vapour.

    This being the case, & water vapour being a much more effective “greenhouse gas” than CO2, it was suggested that increased CO2 was actually a cooling influence.

    That this stuff was disappeared as quickly as it was aired led me to believe it must have some authority, & was definitely nor wanted by the global warming crowd.

    Anyone know more about this?

  10. I don’t think your analysis is faulty as far as it goes. The assumptions for energy and carbon are reasonable. The point about emergent phenomena is most relevant.

    If one considers also that as the CO2 concentration increases it pushes CO2 out of the atmosphere, we can conduct the same exercise:

    About half of all AG CO2 emissions disappears into “the environment”. That is a first order effect. So right at the start, we should deduct 50% from the downstream “heat entrapment” because it literally isn’t there.

    Before any emergent phenomena such as shading from clouds provoked by the puffy tropical clouds I can see outside my window exercise their calming effects, half the putative effect went walkabout.

    Because you are dealing with real numbers and real conditions I suppose this could be viewed as an error of omission. On a first order basis, the answer is half your values, meaning 125 to 200 times the chemical energy released. This is a small fraction of total heating provided by water vapour from irrigation, without which we would all freeze to death. Warm planets are good planets.

  11. The Bern model is used to simulate the evolution of atmospheric CO2 in in the GCMs. In the 1960s the atomic test ban ended the artificial production of C14 and the concentration decayed back to background levels with an e time of 16.5 years so now there is less than 4% of the peak concentration. The Bern model tells us that 40% of the CO2 generated by the Enola Gay’s engines on her trip to Hiroshima is still in the atmosphere and over 25% will still be there in 200 years. Unless the CO2 sinks can sort out isotopes and discriminate in favor of C14 over C12 the Bern model is not correct and the assumption that our emissions will be here for thousands of years is wrong.

  12. Willis, your calculations seem about right. I have calculated it without assuming any loss of CO2 in the first year just to simplify things, and I get nowhere near what the paper says.

    The paper claims about 4x after one year for coal.

    After 1 year, the integrated CO2 radiative forcing (IntFCO2) exceeds the thermal forcing by factors of 3.91, 3.03, and 2.32 for coal, oil, and gas, respectively;

    I only get about 1.7 times assuming no loss of CO2. If 50% of CO2 remains airborne after one year (Bern model?), then on average for a year, 75% is airborne (100+50)/2. So you multiply my “no loss” number of 1.7 times 75% and get 1.275, or about the same as what you got. About 30% more from radiative forcing in first year than there is from burning.

    My calculations:

    Energy from CO2 forcing in one year, assuming no loss of CO2:
    3.7 W/m^2 for each CO2 doubling
    5.10e+14 square meters on Earth
    1.89e+15 watts of power over entire surface of earth for doubling
    8766 hours per year
    1.65e+19 watt-hours of energy in one year, assuming no loss of CO2
    0.0036 megajoules per watt-hour
    5.95e+16 megajoules energy in first year assuming no loss of CO2

    Energy from burning enough coal to produce a doubling:
    1.80e+20 moles of molecules in atmosphere
    Of that 400 out of one million are CO2
    7.20e+16 moles of CO2 in atmosphere
    1.80e+14 moles of CO2 per ppm (assuming 400 ppm of CO2)
    5.04e+16 moles CO2 per doubling (280 ppm x 1.8e+14 moles/ppm)
    0.044 kilograms per mole of CO2
    2.22e+15 kilograms of CO2 per doubling (5e+16 moles x 0.044 kg/mol)
    1.5 kilograms of CO2 / kilogram of coal burned (sub-bituminous)
    1.48e+15 kilograms of coal burned for that doubling
    24 megajoules per kilogram of coal burned (sub-bituminous)
    3.55e+16 megajoules of energy from coal burned for that doubling

    1.68 ratio of energy from forcing in first year divided by energy from burning, assuming no loss of CO2.

    • I think you are off on that 43%, as I am on the 50%.

      According to Roy Spencer, it is about 2.3% per year for whatever is above 295 ppm, or 97.67% remaining.

      https://www.drroyspencer.com/2019/04/a-simple-model-of-the-atmospheric-co2-budget/

      Even so, if you take my “no loss” CO2 number of radiative forcing being 1.7 times the burning, it is still short of what the paper claims (around 4x).

      Using Roy Spencer’s model, you can multiply the 1.7 number by (1/0.0233) to get the total for all the years, which is 1.7 * 43 = 73 times the radiative forcing for each energy unit produced from coal.

  13. I’m going to admit some profound confusion on this one.

    o Effective Black Body Temperature of Earth => 255 K
    o CO2 doubles
    o surface forcing = +3.7w/m2
    o New Effective Black Body Temperature of Earth…. => 255K

    See the problem? The 3.7 w/m2 is not added to the incoming solar energy in the first place. It isn’t “extra” energy being produced in the system. It is a theoretical approximation of the change in the energy flux AT SOME POINT in the atmospheric column. Where exactly that point is, I’ve never been clear on from the IPCC literature or the papers that the IPCC references. I think it is at the Effective Radiating Level, the idea being that everything below that level get warmer and everything above getting colder, and the ERL moving to a higher altitude.

    But once CO2 doubles and the system sorts itself out and comes to a new equilibrium, the watts in and the watts out are exactly the same as before. Mssrs Stefan and Boltzmann would be awful upset to learn otherwise.

    So one could argue I suppose, that the heat capacity of the stuff that gets warmer is larger than the heat capacity of the stuff that gets colder, and so there has been SOME change in the number of joules of energy retained by the planet when equilibrium is re-established. But running the 3.7 w/m2 across the surface of the earth for a 1000 years doesn’t seem to me to be the way to calculate it. Its not a “real” addition to the energy stored in the planetary system.

    • The extra energy comes about from a loss of albedo. Initially the longwave radiation warms the Earth, but that reaches equilibrium and is no longer a factor after a few decades. But meanwhile, it has melted some ice and reduced albedo. More solar radiation is absorbed and less is reflected. This maintains the warming.

      https://www.pnas.org/content/111/47/16700

      • It does? Here’s a quote from YOUR OWN COMMENT ABOVE:

        Energy from CO2 forcing in one year, assuming no loss of CO2:
        3.7 W/m^2 for each CO2 doubling
        5.10e+14 square meters on Earth

        Your own calculation is exactly based on a simple CO2 doubling = 3.7w/m2. When I point out that’s not reasonable you respond with…no…its the albedo change…not the 3.7w/m2… that you just calculated…

  14. Xiaochun Zhang and Ken Caldeira are close to describing the perfect perpetual motion machine ever discovered. 100,000 times ROI for original investment/energy input to the climate system from 1 Kg of coal just seems on the face of it, absolutely preposterous. 1 Kg coal now is equal to 100 Tonnes of coal for thermal AGW forcing over 1000 years? Would be nice to see an abbreviation of their math how they arrive at their conclusion, although I don’t think I want to chase their paper down.

    Even 240-400 times return of excess thermal heat from AGW more than the original thermal heat of the 1 Kg of coal burnt seems to be impossibly high, but if 1000 years is used for the duration, then that precludes any common sense for this argument since we don’t have enough carbon available from all fossil fuels to maintain such high atmospheric levels of CO2 anyway, especially for 1000 years. The only good news out of this is that there is probably some net effect of temporary increases in the average global temps, which on balance is very much net beneficial if we accept how much random natural cooling is much more damaging and deadly than any warming of 1.5- 2 degree C. I’ll take the slight long term warming please.

  15. On the Kenai there have been several 200,000 acre fires in recent years, and now there is a 23,000 acre one actively burning. You not only get CO2, you also get soot and smoke and no pollution control.

  16. Hi Willis – I’ve checked everything as well as I can. All looks OK except the final calculation – the factor by which the cumulative CO2 radiative forcing exceeds the burning of the coal.
    I haven’t looked up the formulae you used (Bern or Joos), but I have used what I believe is a reasonable value of 13 years for the half-life of excess atmospheric CO2.
    The calc goes like this:
    Year 1 : 1.3 times the coal amount (your number)
    Year 2 : 1.2325 (= 1.3 * 0.5^(1/13))
    … etc …
    Year 14 : 0.65 (which is the half-life, ie. 1.3 * (0.5^(1/13))^13
    … etc …
    Year 1000 : 9.57E-24 (= 1.3 * (0.5^(1/13))^999)
    Grand total over 1,000 years : 25

    That’s a lot less than your number.

    Others may have a different half-life for excess atmospheric CO2. For example, Euan Mearns uses a 2.5% decline per year, which is a half-life of about 27 years – https://amedleyofpotpourri.blogspot.com/2017/11/the-half-life-of-co2-in-earths.html
    Using Euan Mearns’ number, the 1,000 year total is 52. Still a lot less than your number.

    Apologies if I have got the calcs wrong, but I’m sure you won’t use them without checking(!).

    • Atmospheric 14CO2 measured after the peak following the 1963 nuclear test ban dropped by 50 percent in 10 years.
      That’s about 5 percent per year.
      The Tau (1/e) decay point is on the same curve at around 16 to 20 years.
      That measurement by itself completely invalidates the idea that CO2 “accumulates” in the atmosphere.
      OCO-2 was designed to resolve anthro-CO2 from natural CO2 fluxes in the atmosphere. It worked as designed. The OCO-2 data show that anthro-CO2 sources are insignificant in the global biogeochemical carbon cycle.

      • “That measurement by itself completely invalidates the idea that CO2 “accumulates” in the atmosphere.”

        That’s what I’m thinking. What better example is there than this 14CO2 test? We actually have real measurements here.

      • Willis – you say “I only get 1.3 times the burning.”. I checked your assumptions and your calculations, and they do indeed give 1.3 times.

        My lower figure for CO2 forcing vs coal burning comes I think solely from my use of a 13-year half-life of excess atmospheric CO2. You can play with the half-life if you like, but I think it would be very difficult to end up with your much higher figure for CO2 / coal. The half-life can be calculated from the observed take-up rate of CO2 by the ocean. This is where my figure of 13 years comes from. (From memory the half-life was actually nearer to 12.5 years, which would reduce my 25 times for CO2 / coal to 24 times – a trivial difference).

  17. Coal combustion 2E20 Joules
    Forcing 1.6E20 Joules.
    Based on 10 milliwatts forcing from 400 to 401 ppm CO2 and 315570 Joules per square meter per year.
    Joules added to the atmosphere do not “accumulate” in the atmosphere.
    CO2 added to the atmosphere does not “accumulate” in the atmosphere.
    Water added to a river does not “accumulate” in the river.

  18. “… Integrating the radiative forcing from zero to infinity yields about 4.5 × 10^10 J of global warming per mol CO2 released to the atmosphere. Combusting one mole of reduced carbon yields about 393.51 kJ/mol (standard enthalpies of formation) [Oxtoby et al., 2011]. Therefore, on a molar basis, the time‐integrated radiative forcing from CO2 released from burning carbon, over its lifetime in the atmosphere, exceeds the thermal energy released by that burning by a factor of about 100,000…”

    Even the weakest coal (lignite/brown coal) has a heat value of 10 MJ/kg http://www.world-nuclear.org/information-library/facts-and-figures/heat-values-of-various-fuels.aspx .

    Assuming the molecular weight is similar to anthracite coal, that’s 6.9 x 10^8 J/mole of coal. Assuming 15 carbons per mole of coal, that’s 4.6 x 10^7 J/mole of C in coal.

    So on a molar basis in C, that ratio would be 978 (assuming “4.5 × 10^10 J of global warming per mol CO2 released to the atmosphere” is correct at 15 carbons per molecule of coal). It would be lower for other types of coal.

    The authors use the standard enthalpy of formation for CO2 as the heat released by the burning of coal. This is simply C + O2 -> CO2. That is incomplete.

    • C + O2 -> CO2 + 8084 Kcal/ Kg of carbon (33940 KJ/Kg)
    • S + O2 -> SO2 + 2224 Kcal/Kg of sulfur (9141 KJ/Kg)
    • 2 H2 + O2 -> 2 H2O + 28922 Kcal/Kg of hydrogen (142670 KJ/Kg)

    In addition to the energy released in those last two equations being ignored…technically the SO2 can have a cooling effect, while H2O as water vapor will have a brief overall warming effect.

    Regardless, this seems to be a very misleading calculation. Using a molar basis of carbon as the comparison and the enthalpy of formation of CO2, burning coal releases as much energy as burning glucose, plant material, or anything else that involves C + O2 -> CO2.

  19. “Clouds are a first order variable regarding the global energy balance. They can move the downwelling energy up and down by hundreds of watts in minutes. That’s on the order of fifty percent of the total five hundred watt average signal. Over ten percent, first order.”

    Indeed. Many climate science communicators describe atmospheric CO2 as earth’s “thermostat”. But that thermostat can’t have a setting for temperature; it’s connected to a furnace with fixed output. So it’s more like a thermostat with a little bit of operational “drift”; after about fifty or a hundred years, the thermostat might cause the furnace to run for one extra second per every hundred seconds. And the thermostat is in a building where clouds spend their nights opening windows and their days closing windows with immediate effect about two orders of magnitude greater than the extra seconds of furnace runtime.

    Doesn’t this issue boil down to a speculative modelling assumption? If one were to assume that condensing greenhouse gas (H2O) can act as earth’s temperature “governor”, then non-condensing GHG’s (CO2, methane,…) could vary over fairly wide ranges without having much effect on temperatures. That’s consistent with what I notice where I live: on a clear cold winter afternoon the ratio of CO2:H2O is at the upper end of its range. That’s when all that extra CO2 we’ve added to the atmosphere has the opportunity to flex its muscle and show me what it can do. And guess what happens as soon as the sun goes down? The temperature proceeds to plunge WAY faster than when it’s cloudy.

  20. Willis
    You wrote
    “Global average downwelling radiation (solar plus longwave) is about five hundred watts. ”

    Would you mind explaining the “five hundred watts” for us less scientific folk because I thought I had seen on this site that it was about 1400 watts -or have I misunderstood some thing ?

    • Thomho: The flux of the sun’s radiation reaching the Earth is about 1365 W/m2.. However, the power delivered to a surface depends on the cosine of the angle at which the radiation intersects with the surface. Lambert’s cosine law. The surface of the Earth that intercepts the sun’s radiation is a hemisphere. The average value of the cosine of the intersection angle integrated over a hemisphere is 0.5. Half of the time one hemisphere is in sunlight and the other half it is in darkness. Therefore, the Earth receives an average of 341 W/m2. About 30% is reflected and not absorbed, bringing the real value down to 240 W/m2.

      GHGs in the atmosphere radiate thermal infrared to space (the only way heat can escape our planet) and towards the ground. The average downward flux of thermal infrared (aka DLR) is about 333 W/m2.

      The AVERAGE total energy delivered is about 573 W/m2.

  21. Willis: I know it is a sin to actually read a paper to understand why it reaches a different conclusion than you do. Both you and the paper calculated different things. Section 3.2 says:

    After 1 year, the integrated CO2 radiative forcing (IntFCO2) exceeds the thermal forcing by factors of 3.91, 3.03, and 2.32 for coal, oil, and gas, respectively; after 100 years, these values increase to 179, 139, and 106 years, respectively; and after 1000 years, they are 1047, 811, and 621, respectively (Figure 1b).

    In other words, the authors think the radiative forcing from burning coal delivers about 4X as much energy to the planet IN ONE YEAR as burning burning the coal, while you have calculated 30% more. The difference arises because you are using 43% as the air-borne fraction for CO2. 43% of the CO2 released by burning over the industrial era still remains in the atmosphere today. The fraction of CO2 remaining from the coal burned in the past year is closer to 100%. They may have based the forcing increase on a change from 280 ppm to 281 ppm, which would increase their forcing by almost 50%. That accounts for most of your disagreement after 1 year.

    However, burning coal adds heat to the climate system in only one year, while the energy delivered by radiative forcing keeps being delivered for as long as the CO2 remains in the atmosphere. How long is that? The IPCC has a model for that, but the slower rate constants in that model are only poorly determined by observations. That model says that about 45% of the average CO2 present over the first 1 year ((179/100)/3.91) will be average CO2 remaining over the first 100 years and the average will be 27% ((1047/1000/3.91) over the first 1000 years.

    If you look at the impulse response function (Equation 2), 22% of emitted CO2 will remain in the atmosphere indefinitely. If the authors integrated over 1 million years, the energy delivered by radiative forcing would be a little more than 200,000 times the energy delivered by forcing for one year. However, looking more closely they actually used equation S2 in the supplemental material. CO2 is cleared slowest from a reservoir equilibrating with the atmosphere with negative exponential term of exp(-t/356,878), meaning 1/eth of the emitted CO2 in that compartment remains after 356878 years, and when this term is integrated to t = infinite, it is still contributing a little forcing after 1 million years! Although the authors provided six significant digit, the rate constants for these slow processes are know with any degree of certainty. Integrating to t = infinity under these circumstances is nonsense, since most of the signal comes from processes that a poorly known. Peer review?

    The more relevant and useful information is in Figure 4 which covers the energy retained by forcing over the industrial era. For coal, that is a little less than 100 times the amount of energy released when the coal was burned. And most of that accumulated energy came from emissions before 1900. Emissions since 1900 have caused accumulation of only 30-fold more energy than was release when it was burned. These values are the same order of magnitude you estimated (250-400 times) over a thousand years.

    So there is no obvious calculation error here, just dramatically different assumptions underlying the calculations. And some of the assumptions in the paper are nonsense.

    (By writing this out so long after your post, I hope I am learning something I wouldn’t have thought on my own without your post.)

  22. Willis wrote: “Here’s the difference. Clouds are a first order variable regarding the global energy balance. They can move the downwelling energy up and down by hundreds of watts in minutes. That’s on the order of fifty percent of the total five hundred watt average signal. Over ten percent, first order.

    CO2 changes, at the other end of the spectrum, are a third order variable. Even an improbable doubling represents a change of less than one percent of the five hundred watt system. Lost in the noise. Counteracted by a small change in cloud emergence time and prevalence.”

    If today is cloudy, the temperature will be about 10 degC cooler than if it is clear. A first-order effect. But that is called weather.

    What happens if climate changes so that we have 1% less cloud cover averaged over 30 years – the traditional period for defining weather. Clouds reflect about 75 W/m2 to space (Rayleigh scattering and surface albedo contribute another 25 W/m2). In that case, a 1% change in cloud fraction is equivalent to a forcing of 0.75 W/m2. That assumption makes clouds slightly less important than CO2.

    What determines what fraction of the sky is covered by clouds on the average? Roy Spenser once commented that when you look at the Earth from space, it is cloudy where the air is rising and clear where it is subsiding. A certain amount of air must go up to remove latent and sensible heat from the surface (that can’t escape fast enough as LWR, OLR-DLR) and the same amount of air must subside. In the long run, the cloud fraction might be relatively constant. However, marine boundary layer clouds aren’t caused by large scale convection.

    Does cloud fraction change with warming? That would make it a feedback – a forcing is a change that is not caused by a change in temperature. The cloud fraction does decrease with seasonal warming. GMST (not the anomaly) is highest during summer in the NH, but that could be due to less evaporation from less ocean in the NH.

    Your first-order and second-order effects were in balance on the climate time scale before we started emitting CO2. The average temperature was high enough that incoming and outgoing energy were in balance. So now only your “third order effects” can cause climate change. Chaotic variations in your first and second order effects certainly can occur, but this is called internal or unforced variability, but these average out on the climate time scale. ENSO is internal or unforced variability, mostly driven by a change in the upwelling of cold water in the Eastern Equatorial Pacific and downwelling of water in the Western Equatorial Pacific. Nothing that I know limits the size of the chaotic fluctuations that produce unforced variability – all we can do it look at the typical variability observed in the last 70 centuries of Holocene climate. Unfortunately that record includes but unforced variability in your first and second order variables and natural forcing. A weaker sun contributed to the LIA. Were the MWP and other warm periods forced by a more active sun (or fewer cosmic rays) or are these examples of unforced variability. Unfortunately, we don’t know whether these warm periods seen in Greenland ice cores were regional, hemispheric or nearly global. They weren’t observed in Antarctica. No large global changes are seen in ocean sediment cores, just slow cooling since the Holocene Climate Optimum. It isn’t clear that unforced variability plus natural variability ever caused fluctuations in GMST as big as or greater than 1 degC. So natural and unforced fluctuations in your first and second order effects are arguably no bigger than and likely much smaller than the warming seen in the past half-century when radiative forcing was growing.

    Which leaves your “third-order” effect from radiative forcing. How big is this effect? A radiative imbalance at the TOA of 1 W/m2/K will warm the atmosphere and a 50 m mixed layer at an initial rate of 0.2 K/year! That is a massive amount of energy for what you correctly call a third order effect. Obviously radiative forcing (2.5 W/m2) and imbalance (0.8 W/m2) are far too big to be ignored in ANY analysis. That initial warming rate is attenuated by heat being transported below the mixed layer and by increased radiative cooling to space associated with warming.

    As best I can tell, your first and second order effects were in balance before radiated forcing began and chaotic fluctuations in these effects have caused only small temperature changes during the Holocene. Which means third order effects are important. A 1 W/m2 imbalance contains a lot of energy even though 500+ W/m2 are entering and leaving the surface all of the time.

    You might think of this as a simple pan balance: You can place a 5 kilogram weight on each side of the pan balance and it won’t tip in either direction. Add a 1 g weight to one pan and the balance tips in that direction. Add a 1 W/m2 imbalance to a balance of 573 W/m2 of energy entering and leaving the surface and the planet starts to warm. You are right, the primary and second order effects come first. However, 70 centuries of Holocene climate suggest chaotic fluctuations in these factors cause only small changes in temperature, like ENSO and the 1920-1945 warm period, and like the 1950-1970 Pause and 1998 (or 2001) to 2013 Pause.

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