Climate Science for Everyone: Why 3% annually is actually a lot of carbon dioxide

Posted on December 29, 2011

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Atmospheric CO2 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 (CO2) 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 CO2 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 CO2 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 CO2 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% CO2 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 CO2 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 CO2 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 CO2 is in the atmosphere since the late 1950s, and as of 2010 the CO2 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 CO2, we’d have gone from an atmosphere that was 0.0285% CO2 to one that was nearly two-thirds CO2. 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 CO2 accumulates.

This whole discussion started because we wanted to know whether or not people were adding “a lot” of CO2 to the atmosphere. From recent measurements, we’re actually adding about 2 ppm per year to the amount of CO2 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 CO2 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 CO2 into the atmosphere at an ever increasing rate. The purple line is the compounding interest line for an interest rate of 0.13%.

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