Climate change math made simple

What's being billed as the U.S. Senate's last chance to pass a bill that deals with climate change, the American Power Act, aims for a now-familiar target: a reduction in greenhouse gas emissions of 83% by 2050. The idea is that if the developed world can manage to reach that goal, the global goal only has to be something like a 50% cut by mid century. As has been pointed out, this will not be easy.

The authors of a recent paper in PNAS call it "a forbidding challenge." Why? It turns out the math and underlying science are much less forbidding. Allow me to take a stab at explaining it in hopes of shedding some light into the science that's driving all the "alarmism." I promise it's not that difficult.

The paper mentioned above, "The Copenhagen Accord for limiting global warming: Criteria, constraints, and available avenues" by Veerabhadran Ramanathan and Yangyang Xu of the The Scripps Institution of Oceanography does an admirable job of reducing the science to something that laypeople can handle. At least, it's more transparent than most similar efforts that pop up in climate journals.

Here are the basics. First, there's the concept of radiative forcing. This is how much extra heat the Earth's atmosphere traps. It's measured in watts per square meter. Many climatologists believe that the planet will warm up (eventually; not right away) about 3/4 of a degree centigrade for every extra watt per square meter of heat that gets trapped. Ramanathan and Xu use 0.8 °C.

Second, there's an equation that's been around for a while that reduces this "climate sensitivity" thing to just a few simple elements:

Hâ² = H₀ ln [(CO_2,E)/CO_2,Ref]

That might look daunting for anyone without college-level math, but it's really not that hard to get a grip on.

Hâ² is the radiative forcing that will lead to no more than a given temperature target, say 2 °C above preindustrial norms.

H_o is a constant with a value of 5.5 . Think of it as the background heat against which the extra radiative forcing is measured.

ln is the natural log.

CO_2,E is the level of carbon dioxide (and other greenhouse gases) in the atmosphere and
CO_2,Ref is the level of carbon dioxide in the atmosphere back before we started burning fossil fuels, or 280 parts per millio

Written this way, it isn't very helpful for politicians and anyone else who wants to know how high CO₂ levels can get before we commit the Earth to warming above a given amount. So some recasting of the equation is necessary:

CO_2,E = (e^Hâ²/H_o) x CO_2,Ref

e is the reverse function of the natural log. It's a constant with a value of about 2.718. So now all you have to do is figure out what Hâ² is -- and know how to use the e key on a calculator. Recall that climatologists like to use 0.8 °C per watt per square meter. Using the 2 °C target, that means Hâ² = 2.5 watts per square meter.

Plugging in the numbers, you get 441 parts per million. And that's why so many experts in the field say we should try to keep things below that level.

But what if the climate sensitivity isn't 0.8 °C per watt per square meter. What if, as James Hansen and colleagues posit, it's more like 1.5? The lower value is basically an estimate of how fast feedbacks such as the amount of water vapour in the atmosphere will effect the global temperature. Hansen et al add in long-term feedbacks, like ice sheet coverage, and come up with the higher number. Use that value and the equation produces:

357 ppm. (Actually, Hansen's math is a bit different, but he comes up with a very similar figure.) And we're already at around 390 ppm.

Of course, this glosses over just how difficult it is to come up with a reliable estimate of climate sensitivity. It is an enormously complex task. But the climatology community is working on it. And the reason members of that community are already recommending we act now to reduce GHG emissions, rather than wait until we've eliminated some of the remaining uncertainties, is that even at the low end of the range of possible sensitivities, we are perilously close to locking in dramatic changes in the climate.