Atmospheric CO

_{2}concentration data from ice core (blue, 1750-1975)

and direct atmospheric measurements (red, 1960-2010) vs. “compounding

interest” model described in post (purple). Click for a larger version.

*In many ways, climate science is difficult. There’s a reason that the best climate models require some of the most powerful supercomputers in the world in order to run. But the most important concepts are easily understood by a non-expert with either a little mathematical skill or the ability to use some simple online tools. This is the inaugural post of a new series that seeks to illustrate how anyone and everyone can understand the most important concepts underlying climate science and the reality that is human-caused climate disruption.*

**Update:** To read other articles in this series, click here.

Are people adding a lot of carbon dioxide (CO_{2}) to the atmosphere? It’s such an easy question to ask, but the answer depends on what you mean by “a lot.” And it depends on what you’re referring to. “A lot” of money to someone who’s broke and living day-to-day could be $10, but “a lot” of money to a millionaire might well be $10,000. Another example is what the World Health Organization considers “a lot” of poisonous arsenic in drinking water: .01 mg/L of water (about ~~.001%~~0.000001% by mass). [correction made as per Nick Barnes’ comment below.]

Generally, most people think of “a lot” in reference to something else, and this is true of the CO_{2} in the atmosphere too. “A lot” of money depends on how much you have to start with, while “a lot” of arsenic in you drinking water depends on an estimate by medical experts of your increased risk of skin cancer due to arsenic exposure. Clearly, in some cases a very small amount of something can still qualify as “a lot.” Atmospheric CO_{2} is one of those cases.

Several weeks back a comment on another of my posts here at S&R claimed that the people were emitting only 3% of the total amount of CO_{2} emitted annually, with natural source responsible for the other 97%. I’ve seen that claim repeated several times recently as well. 3% doesn’t sound like a lot, but maybe, as with the case of arsenic, 3% actually is “a lot.” Thankfully, anyone who has used an interest rate calculator has all the tools he or she needs to determine whether adding 3% CO_{2} every year is a lot or not.

Let’s say that you have a savings account that pays 3% interest annually. If you deposit $100 into the account and wait a year without withdrawing any of it, you’ll have $103. After the second year, you’ll have $106.09. Each year you take the prior year’s number and multiply it by 1.03 (100% +3%) to calculate the value of the savings account at the end of the current year. We can put this into mathematical terms, the equation for the amount in the account after a year is:

After two years, the amount in the account is

We can extend this equation to however many years we want – 5 years would be

We can write this more simply like this:

We can combine all the (100% + 3%) terms into a single term, however, and write it as (100% + 3%)^{5}. And we can generalize the equation for *n* years, any percentage interest rate *r* and any starting balance we want as shown below:

This simple equation is available at any number of online compound interest calculators provided by financial sites, such as these here, here, and here. By all means, play around with them until you’re convinced that my math is correct and that they match both me and each other.

This math represents a simple type of exponential growth – interest compounding in a bank account – but the math is the same for CO_{2} building up in the atmosphere. If every year human activity adds another 3% CO2 to the total, then it adds up, and it adds up pretty fast too.

Scientists have used ice cores from Antarctica to measure how much CO_{2} was in the atmosphere back in 1750, and they estimate it was about 277 ppm (see the blue line in the top figure). Scientists have been directly measuring how much CO_{2} is in the atmosphere since the late 1950s, and as of 2010 the CO_{2} concentration was about 389.78 ppm. That’s an increase of about 113 ppm, or about 40.8%.

Using those interest calculators, enter $277 as the starting balance, 3% as the interest rate, and then play around with the value of n until you figure out how many years it would be before you hit a balance of about $390. You’ll find it’s between 10 and 11 years. Suddenly a 3% annual increase due to people seems like quite a bit, doesn’t it?

The following equation uses the equation above to calculate exactly how many years *n* it takes:

Now, far more than 11 years have passed since 1750 – it’s actually been 261 years. If you could live for 261 years and start with $277 in a savings account that earned only 3% interest annually, you’d have nearly $639,000 in your account by now. That’s a lot more than $390. Put in terms of CO_{2}, we’d have gone from an atmosphere that was 0.0285% CO2 to one that was nearly two-thirds CO_{2}. This hasn’t happened (we’d all be dead if it had), so the rate of growth over that time must have been **much** lower than 3%. We can use the equation below to figure out just how much lower.

When we put in 277 as the starting value, 390 as the present value, and 260 years as how long it took to rise that much, we get that the annual rate is just over 0.13%. That’s about 23 times *smaller* than the 3% that is supposedly a tiny value. By all means, verify this using an online interest calculator – simply enter the rate as 0.13%, the starting value as $277, and the number of years equal to 260 and you’ll see that the ending balance in the account is about $390.

If you have a lot of time to just let money sit in a savings account, you can accumulate a huge amount of money even with a small interest rate because it keeps on multiplying. The same is true of CO2 in the atmosphere. Alas, it’s not a good thing when CO_{2} accumulates.

This whole discussion started because we wanted to know whether or not people were adding “a lot” of CO_{2} to the atmosphere. From recent measurements, we’re actually adding about 2 ppm per year to the amount of CO_{2} in the atmosphere, or about 0.51% per year. (It turns out that the 3% value being bandied about the blogosphere isn’t quite what most people think it is, but that’s a different post). 0.51% about four times more than the 0.13% we calculated from a simple compounding interest calculation.

For anyone who wants a visual representation of what I mean, click on the figure at the top. The actual increase in atmospheric CO_{2} concentration (blue and red lines) was much lower than 0.13% from 1750 to about 1850 (< 0.01%), about the same as the compounding interest rate between 1850 and 1950, and even greater than the compounding interest rate since 1950 (about 0.50%). This means that people are dumping CO_{2} into the atmosphere at an ever increasing rate. The purple line is the compounding interest line for an interest rate of 0.13%.

Categories: Energy, Environment/Nature, Science/Technology

If anyone has requests for what topics you’d like to see explained simply and with a minimum of math that you can’t check yourself, please add your request to the comment thread.

0.01 mg/L is 0.000001%

Thanks for catching that, Nick – I’ve updated the OP accordingly.

Interesting point, but equally interesting is a mechanical question: How do you explain conclusions based on math to the innumerate? The key I think, is something allegedly said to Stephen Hawkins by his publisher, that every equation in the book would cost him ten thousand readers or something like that. In this case, I think the real power of your argument is made in the two analogies–arsenic and compound interest, and would like to see you push on with this series with more analogies and fewer equations, because I think you will reach more people. (Those of us who like equations, already know the ones for compound growth anyway.) You’re making important points and the more people who get them the better.

Now, having said all that, I think the real problem with climate science is the “kick the can down the road” syndrome. People believe, falsely, that we progressives cry wolf too often (Malthus and Club of Rome stuff often cited as examples, incorrectly) because technology will find a way. In other words, even if people buy the graph, they may not act on it for the same reason they dont act on the deficit, social security, automobile efficiency, etc.

Otherwise, I get your point about equations. But part of my goal with this series is to teach some of the basic mathematics as well as the concepts using analogies. Too many people are scared of math, and I’d like to do my small part to reduce the fear. I don’t expect that the majority of my posts in this series will be math based like this one was.

As for the “kick the can down the road” syndrome, I think you’re absolutely correct about that being a big problem. I’ll address that eventually as well, once I’ve come up with a clean way to explain why it’s such a bad idea.

I disagree with Otherwise’s comment that the real power of the argument is in the analogies. The arsenic and the “lot of money to a millionaire” analogies are non sequiturs wrt the CO2 point that is being made and that some people find them convincing is a great illustration of the problems that sensible sceptics face in debating this issue, i.e. too many people have no, or poor, logic. 0.01 mg of arsenic is a lot of arsenic in a litre of water that otherwise has no arsenic in it. The analogy is completely inapplicable to what you are talking about with CO2. A 3% increase in the amount of arsenic in a litre of water might not make any difference (although I wouldn’t know about that , but that is the better analogy). In any case CO2 in the atmosphere is not arsenic in water and that is another reason why the analogy is not really applicable. You also neglect to mention (and I understand that this has no bearing on your mathematical point) that CO2 itself makes up only 3% of all so-called greenhouse gases. The corollary of this is that a 3% increase in CO2 is only actually a 3% x 3%, i.e. a 0.09%, or 0.0009 increase in the gasses that supposedly contribute to warming. On top of this there are a whole lot of other issues which are subjects for other posts, such as how much of the CO2 increase is from humans and how much is natural, and the very big scientific issue of forcings and feedback – the science on this says that the actual CO2 increase has little direct effect on temperature. The disaster-predicting models (which have all been wrong to date btw) base their predictions on the idea that a little bit of CO2 will have an amplifying effect on water vapour, which is 95-97% of greenhouse gases – it is this that cases the problematic warming. Also lately they are saying that increased temperatures will release more methane from the oceans, also adding to greenhouse gases. So, all these other things have to be correct for the science to stack up (even the IPCC reports say that the effects of water vapour are not understood by the scientists – but that does not stop some of them from inserting it into their models as a positive feedback loop and declaring the science “settled”). On the other side, there are studies saying that increased water vapour increases cloud which causes cooling and thus is a negative feedback. If this turns out to actually be the case the whole dangerous human CO2 caused warming argument falls over (because it is so heavily based on water vapour being a positive feedback). So, while your maths seems sound (I did not actually check it because it is simple and I am happy to assume it is correct – in any case the “compounding” point it is making is correct) it does not really prove anything more or add any weight to the already very unsound AGW argument.

Riley, I disagree that my analogies are non sequitors. If you read that section closely, you’ll see that I wrote “…people were emitting only 3% of the total amount of CO2 emitted annually, with natural source responsible for the other 97%.” The comparison to natural sources at this point means that I’m talking about total CO2, not the change in CO2 from one year to the next. A more topical analogy would be to point out that less than 3% of the Earth’s atmosphere is greenhouse gases (it’s actually less than 1%). Regardless, however, it wasn’t a non sequitor.

The actual non sequitor was made by the commenters themselves, who equated a 3% increase with a 3% total concentration.

On a related note, CO2 is more like 5-10% of all greenhouse gases, not 3%, but that’s a minor quibble at this level.

I would ask you to refrain from any further off-topic Gish Gallops in the future, either on this post or on others in the same series. The purpose of this series is to educate people on some of the basics of climate science. Everything in your comment from “You also neglect to mention…” to “…heavily based on water vapour being a positive feedback).” is off-topic and, in such a simple post as this, should have been left out. I’ll ask the admin to moderate out such off-topic ramblings in the future.

Unfortunately, I’d have preferred it if you had verified my math – doing so would have demonstrated a willingness to test my claims yourself, rather than simply trust (or distrust) my conclusions based on preconceived notions. In the future I plan to address most if not all of the many misconceptions you stated in your Gish Gallop, and I hope that you will show a greater willingness to test your own comprehension of climate science than you did this time.

Riley tipped his hand when he tried to slip “sensible skeptics” in there, a clumsy attempt to get you to assign him credibility so that he can better ply an obvious denialist agenda. We like good-faith thinking and debate but we’re not sock puppet theatre and further attempts to game the conversation for corporate or political ends will be dealt with quickly.

To Samuel – don’t jump to conclusions. I wasn’t trying to get anyone to assign me credibility. That should only be gained by what I said. If you are not persuaded, or if you don’t understand it, then so be it – I already realise I am in a minority. What I was doing was acknowledging that there are some crackpots amongst the “deniers” as you call us and also some who “deny” without good reason and I was trying to differentiate between the two, without any specific mention of myself. There is also a large contingent in the equivalent group amongst the “believers” – i.e. “it’s a hot day today, therefore climate change”, or “look, a hurricane, therefore climate change”. I also wasn’t applying any “agenda”. I just put some facts on the table. It ‘s pretty clear that of the two of us, it is you who has the agenda – “corporate or political ends”? You’ve gotta be kidding?

In response to Brian. Geez, I’m sorry for trying to add something. Anyhow, your web page so I will stick to your rules. I had a closer look at your maths and I can confirm that I do in fact agree with it. It seems both correct and openly honest to me. However I still maintain the two things I said in the last post, i.e. 1. despite your protestations, my comment about the arsenic analogy being inapplicable is correct – it is inapplicable, for the reasons I stated previously. Your argument in response to my last post is quite simply wrong. That you don’t get this is “interesting”. and 2, although your maths is correct, it does not prove that “3% annually is actually a lot of carbon dioxide”. However, I may be jumping to conclusions. If the only point you are trying to make by saying “a lot of carbon dioxide” is that if we keep adding 3% it will in a shorter time than many people might think accumulate and eventually double, triple etc then I apologise. However I think that by your use of the words “a lot of carbon dioxide” many if not most people will take this to mean that you are arguing that it matters. Your maths most certainly does not in any way prove that a doubling or even a tripling of CO2 will matter. If you did not mean to make this point then I apologise, but I do think that many who read your article will take it that way.

Riley, adding one or two points to a discussion in an attempt to extend it is acceptable. Adding what I count to be 10 different points is not, especially when most of the 10 points are at best peripheral to the original post.

I must thank you for pointing something out to me, however – I didn’t properly close this post with the conclusions I expected people to get out of it, and I did approach it somewhat sloppily. I’ll take greater care to correct both in the future. So allow me to make the points of the post explicit now.

The points I was making in this post are as follows:

1. Sometimes tiny percentages of things can be bad.

2. Given enough time, even small numbers that compound become big numbers, and then really big numbers.

3. 3% per year CO2 emissions from people sounds small, but due to the effect of compounding, isn’t.

4. 3% per year CO2 emissions from people is actually WAY more than historical values.

That’s pretty much it. I’ll get to the other points in another post.

Brian, your calculations are based on a falsehood. You stated “Several weeks back a comment on another of my posts here at S&R claimed that the people were emitting only 3% of the total amount of CO2 emitted annually, with natural source responsible for the other 97%”. Note that the 3% relates to the annual EMISSIONS of CO2. All of your maths relates to increasing the total amount of CO2 in the atmosphere by 3% every year. I am surprised that no-one else has noticed this before me, unless they saw it & just laughed at you & the commenters who did not scrutinise your words closely enough to see the simple mistake.