simple greenhouse gas models

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Most of randform readers might have heard that the socalled greenhouse effect is one of the main causes of global warming.

The effect is not easy to understand. There are two posts which give a nice intro to the greenhouse effect on Azimuth. One is by Tim van Beek and one is by John Baez.
The greenhouse effect can also be understood in a slightly more quantitative way by looking at an idealized greenhouse model.

In the above diagram I now enhanced this idealized greenhouse model (as of Jan 2017) in order to get an idea about the hypothetical size of the effect of an absorption of non-infrared sunlight and it’s reradiation as infrared light, i.e. the possibly effect size of a certain type of fluorescence.

I sort of felt forced to do this, because at the time of writing (February 2017) the current climate models did not take the absorption of UV and near infrared light in methane (here a possible candidate for that above mentioned hypothetical greenhouse gas) into account and I wanted to get an insight into how important such an omission might be. The simple model here is far from any realistic scenario – in particular no specific absorption lines but just the feature of absorption and reradiation is looked at.

The above diagramm shows the earth temperature in Kelvin as a function of two parameters, as given by this enhanced model. The two parameters can be seen as being (somewhat) proportional to densities of a hypothetical greenhouse gas, which would display this type of fluorescence. That is the parameter x is seen as (somewhat) proportional to the density of that hypothetical greenhouse gas within the atmossphere, while y is (somewhat) proportional to the density near the surface of the earth. Why I wrote “somewhat” in brackets is explained below.

The middle of the “plate” is at x=0, y=0 (please hover over the diagram) which is the “realistic” case of the idealized greenhouse model, i.e. the case where infrared absoptivity is 0.78 and the reflectivity of the earth is 0.3. The main point of this visualization is that linearily increasing x and y in the same way leads to an increase of the temperature. Or in other words, although raising x by a certain amount leads to cooling this effect is easily trumped by raising y by the same amount.

As far as I learned from discussions with climate scientists the omission of non-infrared radiation in the climate models was mostly motivated by the fact that an abpsorption of non-infrared is mostly happening in the upper atmossphere (because methane is quickly rising (but there are also circulations)) and thus leading rather to a global cooling effect than a global warming effect and so it in particular doesn’t contribute to global warming. The enhanced simple model here thus confirms that if absorption is taking place in the upper athmossphere then this leads to cooling. The enhanced model however also displays that the contribution of methane that has not risen, i.e. methane that is close to the earth surface, is to warm upon absorption of non-infrared light and that the effect of warming is much stronger than the cooling effect in the upper athmosphere. Unfortunately I can’t say how much stronger for a given amount of methane, since for assessing this one would need to know more about the actual densities (see also discussion below and the comment about circulations). Nonetheless this is a quite disquieting observation.

I had actually exchanged a couple of emails with Gunnar Myrhe, the lead author of this corresponding chapter in the IPCC report, who confirmed that non-infrared light absorption in methane hasn’t sofar been taken into account, but that some people intended to work on the near-infrared absorption. He didn’t know about the UV absorption that I had found e.g. here (unfortunately my email to Keller-Rudek and Moortgat from 2015 whether there is more data for methane especially in the range 170nm-750nm stayed unanswered) and thanked for pointing it out to him. He appeared to be very busy and as drowning in (a lot of administrative) work, so that I fear that those absorption lines still might not have been looked at. That is also why I decided to publish this now. I sent a copy of this post to Gunnar Myrhe, Zong-Liang Yang and John Baez in June 2017, where I pointed out that:

I have strong concerns that the estimations of the global warming potential of methane need to be better assessed and that the new value might eventually be very different then the current one.

- but I got no answer.

The Wikipedia entry on the Idealized greenhouse model is based on course notes of the course 387H: Physical Climatology by instructor: Zong-Liang Yang where he used this simple model for motivating more complicated models with many layers:

Solution of idealized greenhouse model with emissivity by author Incredio, licence CC BY-SA 3.0

As said, I now enhanced this simple model in a certain way in order to get some insight into the temperature sensitivity of absorption of non-infrared light and it’s conversion into infrared light. I currently don’t have access to a commercial computer algebra system and I sofar haven’t got along with the Sage syntax, so in particular solving spherical Navier-Stokes equations as done in GCM’s is quite out of reach. So I tried to use this enhanced model with Julia. The code is below.
The enhanced model is depicted in the following image:

The notation is as in the Wikipedia article (see first image above), with a few alterations. That is \(S=\frac{1}{4}S_0 = 341 W/m^2\) is here one fourth of the total incoming solar radiation (the factor one fourth is because the area of a sphere (i.e. here the earth) is four times the area of its circular shadow, this is e.g. motivated here) and \(\alpha_p\) is set here \(\alpha_p = \rho_s\) where I chose \(\rho\) as in “reflected”. I kept the notation for the subscripts as they were already used for the temperatures \(T\) in Wikipedia, so the subscripts are \(s\) as in “surface” and \(a\) as in “atmosphere”. The symbol \(\epsilon_{IR}\) denotes the absorptivity/emissitivity of infrared light in the atmossphere (in the Wikipedia entry just \(\epsilon\)), likewise \(\epsilon_{UV}\) denotes the absorptivity/emissitivity of ultraviolet and other noninfrared light, which is here now assumed to be reradiated as infrared light within the atmossphere.

As there seems no “simplify” in Julia, I had to shuffle the algebraic expressions by hand, which is of course error-prone, but I hope there are no mistakes. Below the code and intermediate steps.

Anyways if you look at the code then you see that \(\epsilon_{UV}\) and \(\rho_s\) are dependent on the variables delta1 and delta2. In the model they describe “small deviations” from some standard values. delta1 describes the deviation from the UV absorptivity and delta2 the deviation from the reflectivity of the earth. The idea behind is that if there is some greenhouse gas which absorbs noninfrared and reradiates this as infrared then as delta1 increases the noninfrared absorptivity of the atmosshpere, this is as if there would be “more of that absorbing” greenhouse in the atmossphere. So in the beginning I wrote the word “somewhat” in brackets, because I don’t know the exact relations between absorptivity and density of a greenhouse gas, apart from this I don’t know much about actual densities (see comment about circulation and this post). Likewise delta2 could describe a “more of that greenhouse gas” at the surface of the earth. In the diagram delta1 is x and delta2 is y.

#code is GPL by Nadja Kutz
S= 341.5
deltaAt = 0.0
deltaSur = 0.0
epsuv = 0.0+deltaAt
epsir = 0.78
#epsir=0.78 is corresponding to usual CO2forcing
rhos = 0.3-deltaSur
sigma = 0.00000005670367
Ts= (1/((1-0.5*epsir)*sigma)*(((1-rhos)*(1-epsuv) -0.5*(-epsuv*(1-rhos)-rhos + (1-epsuv)^2*rhos))*S))^0.25
Ta=(1/((epsuv + epsuv*(1-epsuv)*rhos + epsir)*sigma)*((1-(1-epsuv)^2*rhos)*S-(1-epsir)*sigma*Ts^4))^0.25
println(“deltaAt=”,deltaAt,” deltaSur=”,deltaSur,” Ts=”,Ts,” Ta=”,Ta)

#calculation see image Greenhouse.svg
#Term 1
-(1-(1-epsuv)^2*rhos)*S + (epsuv + epsuv*(1-epsuv)*rhos + epsir)*sigma*Ta^4 + (1-epsir)*sigma*Ts^4
#Term 2
(1-rhos)(1-epsuv)*S + (epsuv + epsuv*(1-epsuv)*rhos + epsir)*sigma*Ta^4 – sigma*Ts^4

#Term1 + Term2 !=0
-epsuv (1-rhos)*S -rhos*S + (1-epsuv)^2*rhos*S + 2* (epsuv + epsuv*(1-epsuv)*rhos + epsir)*sigma*Ta^4 + (-epsir)*sigma*Ts^4

#Solve Term1 + Term2 !=0 for Ta
Ta^4= -1/(2*(epsuv + epsuv*(1-epsuv)*rhos + epsir)*sigma)*((-epsuv*(1-rhos)-rhos + (1-epsuv)^2*rhos)*S + (-epsir)*sigma*Ts^4)

#Into Term 2
(1-rhos)(1-epsuv)*S -0.5*((-epsuv*(1-rhos)-rhos + (1-epsuv)^2*rhos)*S + (-epsir)*sigma*Ts^4)- sigma*Ts^4

((1-rhos)(1-epsuv) -0.5*(-epsuv*(1-rhos)-rhos + (1-epsuv)^2*rhos))*S + (0.5*epsir-1)*sigma*Ts^4

#Solve for Ts
Ts= (1/((1-0.5*epsir)*sigma)*(((1-rhos)*(1-epsuv) -0.5*(-epsuv*(1-rhos)-rhos + (1-epsuv)^2*rhos))*S))^0.25

The plot of the function (with some help from Tim) can be got from this code:

function Surfacetemp(deltaAt,deltaSur)
S= 341.5
epsuv = 0.0+deltaAt
epsir = 0.78
#epsir=0.78 wird als CO2forcing angenommen
rhos = 0.3-deltaSur
sigma = 0.00000005670367
(1/((1-0.5*epsir)*sigma)*(((1-rhos)*(1-epsuv) -0.5*(-epsuv*(1-rhos)-rhos + (1-epsuv)^2*rhos))*S))^0.25
#Ta=(1/((epsuv + epsuv*(1-epsuv)*rhos + epsir)*sigma)*((1-(1-epsuv)^2*rhos)*S-(1-epsir)*sigma*Ts^4))^0.25
using Plots


x = y = linspace(-0.1, 0.1, 20)

thanks to Tim for helping me deciphering the Julia documentation

11 Responses to “simple greenhouse gas models”

  1. Jack Webster Says:

    Why isn’t this published on John Baez blog?

    By the way he is currently bashing climate denialists on Twitter:

    …so maybe he is not so happy with what you wrote here.

  2. nad Says:

    Why isn’t this published on John Baez blog?

    I don’t know, I haven’t heard from him in a while. He seems to be very busy teaching category theory.

    By the way he is currently bashing climate denialists on Twitter:

    Yes I saw that and I was already wondering, that he so clearly dismissed Scott Wagners arguments, because I found at least one of them per se not so easily dismissable. And I also didn’t find John Olivers arguments convincing. If Scott Wagner takes money from the oil industry then this makes him less credible, but that doesn’t mean either that all of his arguments need to be wrong. In particular he might be right by accident for some issues.

    That is the earth orbits in a nearly circular orbit around the sun, so there is not much “moving in and out”. But even if there is, I think it is rather clear that Scott Wagner was probably not forgetting about the ellipticity of the earths orbit but that he meant the average distance between earth and sun.The seasons are due to the changing angle of the earths axis with respect to the orbit plane.

    The earth might at some point crash into the sun, because it is loosing energy but then the sun is probably loosing quite a bit of mass. I have quickly looked for concrete measurements of the earths distance to the sun but I haven’t sofar found anything. I imagine that if there would be big alterations we would have heard of that, but who knows….after looking a bit at the state of science and media I wouldn’t wonder too much if not. So I find this argument actually not so easily dismissable and per se a rather important point that apriori needs to be checked.

    What about the heating by humans? The earth has according to Wikipedia a surface of $latex 510*10^{15} m^2$, multiplied by the above average power $latex 341 kW/m^2$ this is about $latex 174*10^{18} W$. If a human would convert all eaten energy into heat than as I had calculated here this would be for 1500kcal about 1.743 KWh per day, i.e. $latex 1743 Wh/24h=72.625W $ There are currently $latex 7.6*10^9$ people on earth so this is $latex 72.625W*7.6*10^9=551*10^9W $ so roughly about a factor of $latex 10^9$, i.e. one billion off from from the sun’s average power on earth. So this appears indeed rather small, apart from the fact that that one has to compare this with the “heat” that would be produced via the decay of food.

    So I dont have no time to do bigger calculations but I think the earths orbit is something that has to be looked at in principle, the heat produced by humans is something I would leave out in calculations for the moment.

    If mankind would need to alter the earths orbit then this will need a lot of science and I am not sure if “society” is doing its best here.

  3. Jack Webster Says:

    John Baez seems to implicitly comment on your remark with the correction that the assumption of 1500 kcal per person is way too small – he is right – a human needs about 2500 kcal per day and thus 115 Watts and not 72 as you calculated. He furthermore discusses that people use about 475 Watts of electricity, so this is about 4 times the amount of power you considered negligable in contrast to the sun, but then climate scientists freak out about electricity consumption and not about human food. You contradict yourself.

  4. nad Says:

    I doubt that he commented on what I wrote, since he told me that he reads this blog not very often.

    The 1500 kcal is what is probably to be expected as a future food average. I thought that human heat doesn’t play a big role because it is mostly reradiated, it basically enters the above balance. It is the accululation of gases in the athmossphere that changes the temperature. Thus what may play a role is actually rather the CO_2 contributions of humans.

    Let’s do a quick estimation:

    Sorry I have only found in the german Wikipedia Gasaustausch an info about the conversion of O_2 into CO_2:

    Atemzeitvolumen des Organismus (aktive äußere Atmung). Bei normaler Atmung in Ruhe werden vom Menschen pro Liter Atemluft ungefähr 170 ml Sauerstoff eingeatmet und 130 ml wieder abgeatmet.

    which says that of 1 liter air approx. 170 ml of O_2 is inhaled and 130ml exhaled. So about 17% of inhaled air is O_2 and 13% is re-exhaled. That is a bit off of what the english Wikipedia says about the average air:

    There is 78,084% N_2 and 20.946 % O_2 and 0.0407 % CO2 in the air

    , but this might be due to the different air pressures at different heights. So 70mL=0.07L of the 170 mL inhaled oxygen is converted and I assume it is probably mostly converted into CO_2 that is 70/170= 41%. The air volume per minute is in the english wikipedia 6 L/min, in the german it is 7.6 L/min, so lets take 7 l per minute, so in a lifetime of 70 years this is 0.07L*7*60*24*365*70=18028080L according to google calculator, so about 18 million Liter O_2 is converted into CO_2 per person and lifetime. So the current 7.6 billion people will convert 18*7.6*10^15 L= 14*10^16L.

    Unfortunately I couldnt find any numbers about the oxygen volume on earth, so lets assume the relevant atmossphere is 15 kms thick. The earth radius is 6371 kms that gives a volume of (6371)^3*3.14*4/3 kms^3 adding 11 km to the radius and deducing the earth volume and taking 20 % oxygen thus gives:
    ((6382)^3-(6371)^3)*3.14*4/3*0.2 km^3= 1123510552.79 km^3 =
    1.12* 10^9 km^3=1.12*10^9*10^27m^3=1.12*10^9*10^27*10^3L= 1.12*10^(9+27+3)L=1.12*10^39L So roughly if there would be no 02 production (like by plants) there would be an oxygen volume for way less (I dont know the lethal proportion) than 10^39/10^16 =10^23 generations of humans. So at the moment this doesn’t look dramatic.

    But what about the CO2?
    Taking a factor of 0.000407 for the fraction of CO_2 gives 2286343.97 L= 2.3*10^6 L So human breath conversion of 14*10^16L might contribute significantly to CO_2 percentages.

    Please take the calculations in this comment here with utmost caution, as I didn’t check them and it is very likely that I missed a factor here and there. This calculation is just for roughly estimating whether it might be relevant to look into this further and I currently think yes.

  5. cindy Says:

    It is the accululation of gases in the athmossphere that changes the temperature. Thus what may play a role is actually rather the CO_2 contributions of humans.

    What is accululation?
    This is ridiculous. You shuffle around some numbers and brackets and cryptic ^ and chemical abbreviations and you think you can display some magic which tells us that human breath might be dangerous or what?

  6. nad Says:


    accululation is a missprint -it should read accumulation. I was in a bit of a hurry, when I wrote the comment above. That is socalled greenhouse gases are stored and again stored (I.e. accumulated) in the air and sofar nobody removed those gases and those gases seem to be one reason why solar light is leading to more heat up on earth.

    I didn’t say that human breath is dangerous, I just said that by the above calculations that human breath implicitly might contribute to rather fastly rising temperatures on earth due to the accumulation of the CO_2 in breath in air. One in particular has to compare this to the decay of plants, which also produce CO_2 if they are not eaten by humans or animals.

    You can see this “number shuffling” as a magic if you want, it is though a “magic” that has been developped by very careful observation of natural laws and a lot of humans trying to make sense of that. It has sofar taken people to the moon, so it hasn’t been too unsuccessful in the past, in fact science had sofar been much more successful than any other attempt in doing “magic”, i.e. in making people “do” things that were quite unthinkable once. But of course this is no guarantee for the future. Physical laws can change any second and then a lot of physicists and mathematicians are pretty useless. I keep saying this on this blog.
    The ^ means “raised to the power of”, so 10^2 is 10*10=100 and 10^3=10*10*10=1000. The equality sign means that all those expressions are considered to be the same “thing”. The brackets are an abreviation, which makes use of the distributive law between multiplication and addition and for writing sums in exponentials. The abbreviations for chemicals are explained in Wikipedia. CO_2 is carbon dioxide.

  7. Dumuzid Says:

    It is not only plant decay that makes plant produce CO2 but you in particular forgot to mention soil respiration.

  8. Dumuzid Says:

    nad wrote:

    So at the moment this doesn’t look dramatic.

    So why do you then write posts about oxygen recesseion?

  9. nad Says:

    Thanks for pointing out the soil repiration link.
    I wrote this mainly because there is still a big air volume, but then things get thighter.
    OK let’s look for example at Hypercapnia.
    The german Wikipedia writes:

    Bei einer Konzentration von 1,5 % (15000 ppm) nimmt das Atemzeitvolumen um mehr als 40 % zu.

    From a concentration of 1.5% the respiratory minute volume increases by more than 40%, no citation at that concentration the english Wikipedia writes there (and cites Lambertsen, Christian J. ( “Carbon Dioxide Tolerance and Toxicity”. ) is to be expected a “mild respiratory stimulation ” after an exposure for longer than a month. An immediate carbon dioxide poisoning seems to take place in both Wikipedias at around 8%. The effects of higher dosis and long term exposure seem to be more difficult to establish, in particular it seems that Hypercapnia doesn’t follow the usual Haber’s rule. Google calculator says that: 0.08/0.0004=200. So if one takes the above CO2 estimate than about 200* 2.3*10^6 L = 4.6*10^8L of CO2 in the air will make air rather poisonous. So if the above estimation of 14*10^16L is right than humans will exhale about 3*10*8=300.000.000 times more CO2 than that poisonous dose….but of course there are still plants producing oxygen….

  10. Dumuzid Says:

    There is the article Global land change from 1982 to 2016 which says that

    We show that—contrary to the prevailing view that forest area has declined globally5—tree cover has increased by 2.24 million km2 (+7.1% relative to the 1982 level).

    So this suggest that plants produce more oxygen!

    This by the way has also been reported in a german mass media outle (since you were suspecting the media might not report):

    Are you sure that your estimations are at all useful? You write:

    Please take the calculations in this comment here with utmost caution, as I didn’t check them and it is very likely that I missed a factor here and there. This calculation is just for roughly estimating whether it might be relevant to look into this further and I currently think yes.

    But if you say this is shaky then maybe it is even too shaky to conclude that one should “look further into this”.

  11. nad Says:

    Thanks for pointing out the article.
    Another effect of those calculations is to see how long it takes to do them.
    And it is not only the calculaitons themselves but also the assumptions that have been made here that are shaky, like in particular the absolute air volume.

    But anyways as we know the biggest CO2 production is via fossil fuel usage, so let’s compare human breath to global fossil fuel CO2 production.

    It is written in here that:

    Global energy-related CO2 emissions
    rose by 1.4% in 2017, an increase of 460 million
    tonnes (Mt), and reached a historic high of 32.5 Gt.

    What is the volume of one ton CO2? Wikipedia says that at 0°C and a pressure of 1013 hPa CO2 has a density of 1.98 kg/m^3 so roughly this gives 1000kg/1.98kg m^3= 505.0 m^3.
    This is in the same range as the 556.2 m^3 that has been found here for 25 °C. So assume 556m^3 is correct then this is 32.5*10^9*556*1000L= 1.8*10^16L We had for human breath 14*10^16L/70 CO2 volume in a year that is 0.2*10^16 L. So annual CO2 from fossil fuel emissions are about 9 times more than human breath exhalations or human breath is about 11% the size of global CO2 fossil fuel emissions or 0.2/(0.2+1.8) =0.1, i.e. 10% of fossil fuel and human breath emissions together.

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