Understanding Global Warming Scale Issues

Aside from the endless stream of Cantor cranks, the next biggest category of emails I get is from climate “skeptics”. They all ask pretty much the same question. For example, here’s one I received today:

My personal analysis, and natural sceptisism tells me, that there are something fundamentally wrong with the entire warming theory when it comes to the CO2.

If a gas in the atmosphere increase from 0.03 to 0.04… that just cant be a significant parameter, can it?

I generally ignore it, because… let’s face it, the majority of people who ask this question aren’t looking for a real answer. But this one was much more polite and reasonable than most, so I decided to answer it. And once I went to the trouble of writing a response, I figured that I might as well turn it into a post as well.

The current figures – you can find them in a variety of places from wikipedia to the US NOAA – are that the atmosphere CO2 has changed from around 280 parts per million in 1850 to 400 parts per million today.

Why can’t that be a significant parameter?

There’s a couple of things to understand to grasp global warming: how much energy carbon dioxide can trap in the atmosphere, and hom much carbon dioxide there actually is in the atmosphere. Put those two facts together, and you realize that we’re talking about a massive quantity of carbon dioxide trapping a massive amount of energy.

The problem is scale. Humans notoriously have a really hard time wrapping our heads around scale. When numbers get big enough, we aren’t able to really grasp them intuitively and understand what they mean. The difference between two numbers like 300 and 400ppm is tiny, we can’t really grasp how in could be significant, because we aren’t good at taking that small difference, and realizing just how ridiculously large it actually is.

If you actually look at the math behind the greenhouse effect, you find that some gasses are very effective at trapping heat. The earth is only habitable because of the carbon dioxide in the atmosphere – without it, earth would be too cold for life. Small amounts of it provide enough heat-trapping effect to move us from a frozen rock to the world we have. Increasing the quantity of it increases the amount of heat it can trap.

Let’s think about what the difference between 280 and 400 parts per million actually means at the scale of earth’s atmosphere. You hear a number like 400ppm – that’s 4 one-hundreds of one percent – that seems like nothing, right? How could that have such a massive effect?!

But like so many other mathematical things, you need to put that number into the appropriate scale. The earths atmosphere masses roughly 5 times 10^21 grams. 400ppm of that scales to 2 times 10^18 grams of carbon dioxide. That’s 2 billion trillion kilograms of CO2. Compared to 100 years ago, that’s about 800 million trillion kilograms of carbon dioxide added to the atmosphere over the last hundred years. That’s a really, really massive quantity of carbon dioxide! scaled to the number of particles, that’s something around 10^40th (plus or minus a couple of powers of ten – at this scale, who cares?) additional molecules of carbon dioxide in the atmosphere. It’s a very small percentage, but it’s a huge quantity.

When you talk about trapping heat, you also have to remember that there’s scaling issues there, too. We’re not talking about adding 100 degrees to the earths temperature. It’s a massive increase in the quantity of energy in the atmosphere, but because the atmosphere is so large, it doesn’t look like much: just a couple of degrees. That can be very deceptive – 5 degrees celsius isn’t a huge temperature difference. But if you think of the quantity of extra energy that’s being absorbed by the atmosphere to produce that difference, it’s pretty damned huge. It doesn’t necessarily look like all that much when you see it stated at 2 degrees celsius – but if you think of it terms of the quantity of additional energy being trapped by the atmosphere, it’s very significant.

Calculating just how much energy a molecule of CO2 can absorb is a lot trickier than calculating the mass-change of the quantity of CO2 in the atmosphere. It’s a complicated phenomenon which involves a lot of different factors – how much infrared is absorbed by an atom, how quickly that energy gets distributed into the other molecules that it interacts with… I’m not going to go into detail on that. There’s a ton of places, like here, where you can look up a detailed explanation. But when you consider the scale issues, it should be clear that there’s a pretty damned massive increase in the capacity to absorb energy in a small percentage-wise increase in the quantity of CO2.

13 thoughts on “Understanding Global Warming Scale Issues”

  1. A crucial gas in warming the Earth went up by 35% (from 0.03% to 0.04%), and the surface temperature of the Earth went up 2 K, from something like 289 K to 291 K, or 0.7%. In the future, CO2 is estimated to hit 0.08% or higher, which could mean 2 or 3K or more, another 1 or 2% rise in planetary temperatures. So CO2 amounts have a disproportionate but ultimately small impact on world temperature; unfortunately, most life on this planet including us is pretty sensitive to that temperature.

  2. I think that converting to absolute numbers makes the problem worse. You just changed the units and scale from something that is unimaginably small to one that is unimaginably large. Both are unimaginable and neither helps in comprehension.

    I think David Starner’s comment is spot on. Converting to percentages gets you to a scale that makes sense to human beings. A 35% change is pretty big. That such a large change in CO2 leads to a small percentage change in temperature seems pretty reasonable. This gets us past the “small change leads to big change” problem when we realize that it is just the opposite.

    Of course, there was also the less intuitive and thus less persuasive approach of explaining how very minor changes in input to a function can lead to drastic changes in output. That would be the more mathematical approach and in keeping with your theme.

    1. It’s always hard to figure out how to address problems like this. I thought that in this case, showing that something that seemed to small to matter was also too big to wrap your head around was a good approach. But, as usual, YMMV.

    2. I agree with blu28’s comments. The general public are terrible at understanding very large absolute numbers, so converting from relative to absolute amounts only heightens misunderstanding. The key observation is that CO2 levels have increased by 35%, a large amount, over about 150 years, an extremely short time compared to previous changes in CO2 levels of similar magnitude in the earth’s history.

      As to comment that 300 ppm can’t possibly be significant, maybe you could suggest the the skeptic spend some time in a room with 300 ppm of HCN in the atmosphere (after all, that can’t possible be significant!).

  3. Fact, global temp is rising!
    Fact, the rise is due to human activity!
    Fact, sea levels are rising!

    Unknown, how bad will it be?

    This is from NASA website>>>

    “Given what we know now about how the ocean expands as it warms and how ice sheets and glaciers are adding water to the seas, it’s pretty certain we are locked into at least 3 feet of sea level rise, and probably more,” said Steve Nerem of the University of Colorado, Boulder, and lead of the Sea Level Change Team. “But we don’t know whether it will happen within a century or somewhat longer.”

    Three feet of sea level rise, over 100 years, will displace a lot of people, but it will not be as bad as the abandonment that we have seen in the "American Rust Belt" due to job loses in those areas over the past 60 years.

    So, until NASA raises the threat level by several notches, you can call me a "climate denier".


    1. So Ivan, your argument is two-pronged, if I’m not mistaken:

      1) Climate change will only affect us through sea level rises and through no other changes (e.g. we will not be directly or indirectly affected by the heat which melts the polar icecaps and expands the oceans except through that change in the mean ocean level)
      2) this single effect of climate change, the sea level rise, will be distributed evenly across the globe, and therefore only of interest by people who are 3 feet above the current extremes of sea level behaviour.

      Is that right?

  4. So, the “sceptics” asks:
    “something wrong with warming theory in relation to co2 as a parameter?”.
    Markcc answers:
    “There has been added an unimaginable amount of co2 to the atmosphere. More than you can understand, because you can’t grasp how big the atmosphere is.”. Nothing about warming theory.

    I dont really see an answer to the question there. So what if the earth is big and holds alot of atmosphere?
    If the earth and atmosphere were twice as big, i wouldn’t have a problem with twice the co2.

    I can come up with huge numbers myself, while addressing the original question.
    Greening of the earth, as a result of higher co2, yields enourmes amounts of extra vegitation.
    I’m not a “mathematician”, so i wonder how that number would compare to the enormous amount of extra co2. Greening sounds like a lot of food to the world. Sounds like it would decrease co2 levels, too. Your warming crisis seems less of a problem.

    How about the fact the people taking driver licenses today, have not lived on a warming earth, despite the co2 level? Is it greening?

    I think what the “sceptic ” is really asking about, is: “Is there really a problem here with human made co2 and life on the planet?”.

    I don’t see any difference. Sea level is rising at the same rate as 300 years ago. Nothing about the weather is worse today, than 200 years ago. It’s just mildly warmer with more benefit than disaster. So far, all we have is doomsday alarmism.

    I know they say that for example 80% of australias coral reef is dying, but it simply isn’t true. There is still the same diving tourism, and they dont see 80% missing. They see 5%, despite the official 80%. But they’re not climate scientists, I gather.

    Temperature, according to statistics, have risen a little the past 200 years. We’re talking about less than 1c. How precise is that, when for example Antarctica has only 7 measurement stations?

    It seems to me that converting the numbers to absolute scale confuses the issue.
    Telling people that co2 has increased 35%, instead of ppm, is to give them less information. No context. They’re left with the impression that we’re 35% worse off, than we were before, which definatly isn’t the case.

    I think this is all about confusion of motives. I think the sceptic has realistic conserns about the “climate disaster” being bullshit, while Markcc accuratly claims that more co2 equeals more heat.
    I think the sceptic knows the earth is warming. He’s probably aware of the 97% concensus among scientists who agree that humans contribute to heat.
    Well, how can any person answer no to the question? Can a person even fart without making a contribution to added co2 and heat? Nobody dissagrees with that. I don’t think anyone thinks that the climate is static.
    I think they just dissagree about all the alarmism, dissaster scenarios, uncrontrollable heat and doomstay, which you guys seem to be spreading.

    The truth is, that if the earth was cooling, we’d have no chance stopping the cooling with more co2.

    Yes, there is a problem with the warming theory. The earth has not warmed as much the last 20 years, as the IPPC projections suggested. Theory is off.

    1. While you have a couple of good points, you data is way off.
      Consider that excess CO2 only leads to increased vegetation in locations where the lack of CO2 is the bottleneck that prevents greater vegetation.
      In the early part of the 20th century the sea level was rising at 1.6 mm per year, while it is currently rising at 3 mm per year.
      The IPCC had a range of models and the actual warming had been well within that range.
      Whether the climate is ultimately going to better or worse is very subjective. What is undeniable though is that the more rapid the change the worse things will be as the world reacts to the change.
      Presenting the change as 35% is clearly not the best way either. It tells us more about the change in CO2, but what we want to represent to the public is the impact, and that is very difficult to represent by a single number. Perhaps global cost in dollars or global deaths would be more suitable.

  5. There is a classic paper 2007 on this by John Sterman and Linda Booth Sweeney Understanding public complacency about climate
    change: adults’ mental models of climate change violate
    conservation of matter.

    Their argument is basically, that it’s not so much the *scale* of the numbers as that people, even very smart people, aren’t great at reasoning about changing the rate at which things change, which is at the core of climate change models – (and, for that matter, the difficulty of cooking a decent roast chicken from first principles.)

    MIT news did a layperson’s summary on this, and it became a very nice infographic at National Geographic, which is a recommend intro to the non-obvious behavior of these systems.

  6. “2 times 10^18 grams of carbon dioxide. That’s 2 billion trillion kilograms of CO2.”

    Is that right? Seems to me it’s 2 thousand trillion kilograms. I know “trillion” used to have a different meaning, at least to the Brits, but that was bigger, not smaller. How about “2 million gigatonnes?”

  7. “If you actually look at the math behind the greenhouse effect, you find that some gasses are very effective at trapping heat.”

    “Trapping heat”, and by that you mean “heat up”?

    The effect of the heat absorbing gases is measured by satellites and averaged to a mean decrease of a couple of Watts from the increased amount of heat absorber in the troposphere.

    It is the decrease in emitted intensity from the troposphere that is the effect of greenhouse gases. If the emitted intensity has decreased, the temperature of the troposphere has decreased. The effect of colder air on surface temperature is without a doubt a lower surface temperature.

    The concept of radiative imbalance claims decreased intensity of the troposphere causes the opposite of what physics say about temperature and decreased temperature. If intensity decrease, the energy content has decreased, and the only possible effect on anything in the surroundings is a lower temperature.

    When solar energy is considered to be somewhat constant and you increase the amount of heat absorber in the already much colder gas layer, the same amount of energy is absorbed by more molecules resulting in less energy per molecule, which is the definition of lower temperature.

    “The earth is only habitable because of the carbon dioxide in the atmosphere – without it, earth would be too cold for life.”

    You imply that if there was no atmosphere and the surface recieved radiation at an intensity between 680 and 1360W/m^2 over the irradiated hemisphere, the surface would be colder than it is when it sits under a filter of air and clouds at an average temperature of -18C?

    How does lower intensity radiation and cold air lead to a higher temperature than high intensity unfiltered radiation?

    That is the opposite of what we know about radiating bodies.

    When visualizing it, imagine a spherical stone in space heated by radiation of the raw solar constant over a hemisphere, and then you add a layer of gas around the heated sphere, with a mean temperature of -18C.

    Where do you find extra energy added to the surface? All I can see is less energy to the surface in order to heat both the surface and the cold gas layer.

    If there is fixed input to the system, adding a heat absorbing gas at a low temperature can never increase the temperature.

    When input is constant and heat absorbing molecules is increased in number, we can be absolutely certain that the temperature will decrease. This will show as lower radiated intensity from the troposphere. The “radiative imbalance” of a couple of Watts is not a sign of “trapped heat”. When something heats up the intensity increase. Always, without exceptions.

    The radiative imbalance is what you get from increasing the amount of dry ice in the atmosphere.As can be seen in the spectrum from satellite measurment, co2 affect radiation heavily by decreasing the intensity in a narrow band, keeping the intensity of those wavelengths firmly at the equivalent of a blackbody at ~215K.

    Co2 in the troposphere has it´s fingerprint in the spectrum where the intensity of radiation is equal to that of dry ice.

    If the spectrum shows how co2 lowers the intensity of radiation, co2 doesn´t warm anything up.

    “Small amounts of it provide enough heat-trapping effect to move us from a frozen rock to the world we have. Increasing the quantity of it increases the amount of heat it can trap”

    Saying that an increasing amount of cold heat absorber in a system heated by an external source, would increase temperature by “trapping heat” in a gas at a mean temperature 33C lower than the surface, is like saying:

    -Let´s light a fire with these ice-cubes and then we build a flame-thrower with that dry-ice fire-extinguisher.

    I would like to ask you to check the litterature for evidence of co2 having a temperature increasing effect on its own heat source. There are lots of studies on co2:s absorbing properties, but I have not found anything to support that claim. Heat absorption in a colder gas does not in any way mean that the heat source increase its emitted intensity.

    When considering radiative heat transfer, the relationship is the opposite.The rate of transfer is equal to the difference in temperature. Absorption is greater the lower the temperature gets. According to the well established concept of radiative heat transfer, if absorption increase in the lower temperature body, it is inevitably an observation of decreasing temperature of the absorber. If co2 increase heat absorption in the colder atmosphere, it lowers temperature. This we know for certain from basic physics.

    Global warming from increasing co2 needs to explain how decreasing intensity from increasing the amount of heat absorbing mass while energy input is constant, relates to the fact that the theory of radiative heat transfer say the exact opposite.

  8. Good attempt. I agree with Blu28 and others that converting to huge absolute numbers confused people more than enlightens them.

    However, the shift to Kelvin is very relevant. As Dan illustrates, 2 degrees K increase is less than 1%, not a big change before we get into trouble.

    Now, how to explain that Absolute temperature is the more relevant measure here? Black bodies? (Why is everryone leaving the room?)

    1. I think there’s a trivial way to explain why Absolute temperature is more relevant – “space” around Earth is at roughly 0 K – so if we’re moving from a mean surface temperature of 290K to a mean of 292K, that’s less than 1% change, but it’s a shift from a mean of 17 °C to 19 °C.

Leave a Reply