Category Archives: Big Numbers

Understanding Global Warming Scale Issues

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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.

Big Numbers and the Lost Plane

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(In the original version of this post, there were two places where I wrote “cubic” instead of “square”. It was a stupid mistake, but just a stupid wording mistake. The calculations are unchanged: I was calculating square units, not cubic – I just typed the wrong thing.)

I haven’t written about this in a while, but for a long time, I’ve been fascinated by just how bad humans are at actually understanding big numbers. When it comes down to it, we’re comfortable with numbers we can count on our fingers. The further you get from our fingers, the worse we get at it.

We see this over and over. I’ve been noticing a new variation on this lately, in discussions about the missing Malaysian airliner. For example, I was at the dentist tuesday morning for a tmj mouth guard because I’ve been told that I grind my teeth at night. They’ve installed TVs in the exam rooms, to give you something to do while you wait for the novocaine to kick in, and they had it turned to CNN. The technician who was setting things up was talking to me about what they were saying, and she kept repeating: it’s a plane! It’s huge! Why can’t they find it?

If you think intuitively, it’s a good question!

In fact, if you look at it with basic intuition, it seems insane. A Boeing 777 is a huge plane! The thing is nearly as long as football field, and it’s length wingtip to wingtip is just short of 200 feet. Fully loaded with fuel at takeoff, it weighs more than 300 tons! How can you lose something that large? Even if it broke into pieces, the debris field from it would need to be huge, and it must be easy to find!

But that’s relying on intuition, and as I said in the introduction, our intuitions scale very poorly. To show what I mean, I’m going to work through a process that will hopefully help make it clear just how badly our intuition fails us here.

We’ll start by coming up with an estimate of the quantity of debris.

Assume that the fuselage is a continuous tube. The official stats say that the diameter of the fuselage is 20 feet, and it’s 242 feet long. So a deliberately overlarge estimate of the total surface area of the fuselage is 24220π – or about 15,000 square feet. Assume that the surface of the wings is about he same, and you get a total surface area of about 30,000 square feet. That means that if the entire body of the plane was peeled and laid flat, and it all floated on the water, it would cover 30,000 square feet of the surface. (In reality, much of the plane would sink, but the seats would float; on the whole, it’s probably a wash; the 30,000 square feet is probably still order-of-magnitude correct for the amount of debris visible on the surface.) Sounds like a whole lot, right?

To get an idea of how easy that is to find, we need to consider how much area we need to search. From what we know about how long the plane could have been in the air, and what direction it could have been moving, we’ve got a search area that’s at least a couple of thousand square miles.

One square mile is 27878400 square feet. So assuming that the plane crashed, and its wreckage is distributed over one square mile, that would mean that in the crash area, if every bit of the plate floated, just over one tenth of one percent of that square mile is covered in wreckage. That’s piling together very unrealistic assumptions – a highly constrained debris field, all of it floating – in order to stack the odds of finding it!

We’re using some optimistic assumptions here – but even so, even with a debris field that’s highly (unrealistically) constrained, it’s still very sparse. (In fact, it’s likely that if the plane crashed in the water, the debris field is spread over more than one square mile.) That’s what we’re looking for: an area where less than one tenth of one percent of the surface is covered, in an area covering several thousand square miles.

A thousand square miles is pretty squarely in the zone of stuff that we don’t really understand very well. Getting an understanding of just how big an area a thousand square miles is… it’s not something that our intuition will help with. But statistically, we’re looking for a patch of debris where the visible artifacts cover less than one ten-millions of one percent of the area being searched. If the search area were a football field, you’d be looking for a little fleck of tinfoil, 1/32nd of an inch on each side.

Only even that is making it too easy: it’s not one piece of tinfoil, 1/32nd of an inch on a side. It’s a fleck of foil which was shredded into a thousand even tinier pieces.

That’s assuming a search area of only one thousand square miles. But the search area is, potentially, quite a lot larger than that. Searching for the tinfoil on a football field is easy in comparison.

Once you understand how small an airplane is in comparison to a small patch of ocean, and you understand just how big a small patch of ocean is, then it’s not at all surprising that we haven’t found it yet.