I am good at math. Way gooder than I am at writing. In fact, I have always been irritatingly good at math. In second grade we played Around the World with math flash cards, and I crushed it. One day I went around the room four times and never lost. It’s a complete mystery why I never found myself duct-taped to a telephone pole. Probably because everybody knew I wasn’t that good at anything else. Yeah, I love math.

Perhaps the thing I love the most about math is it’s unforgiving rigidity. Two plus two has only one answer, no matter what you do. But I also love math’s utility in my world. I have a budget spreadsheet filled out to my retirement. I use math to keep track of every hockey roster I assemble. I even use math to track whether I’m going to keep all my perks on Southwest as the end of the year approaches. And I love the purity of never being able to blame math for anything. If something gets miscalculated, I’m the one who did it.

I love science just as much as I love math. It’s not as easy to say I’m good at science, because that could mean a number of different things. My degrees are all in what the University of Colorado calls Aerospace Engineering Sciences, and I’ve taken some good science classes, and the running joke whenever I’m introduced to someone new is that I’m a rocket scientist. But even though I love the scientific method and try to apply it wherever I can in life, I don’t consider myself a scientist. I’m an engineer – a mercenary who exploits science and math for profit.

I love models just as much as I love science and math. Now, when I said I love math and science, there was probably no ambiguity. But when I say I love models, well, what does that mean? Models that walk down a runway? I’m sure they’re nice people, but no, that’s not what I mean. Model airplanes? My brother loved those, and yes they are pretty cool, but I never had the patience to work with them the right way. Modelo beer? Well, yes, but that’s not where I was going with this.

I love mathematical models. And what the hell are those? Just like a model airplane is an attempt to capture the most important physical aspects of a real airplane, a mathematical model is an attempt to capture the most important physics of a real “whatever”. That “whatever” could be the Earth’s atmosphere, or it could be your retirement planning, or it could be the quantum mechanical foundations of everything we see around us. All three of these examples have some fairly complex mathematics behind the models, and all three of them also have to make assumptions where there isn’t a reasonable mathematical building block.

Models don’t have to be complicated. For example, suppose you are in your car, and you want to get from point A to point B. Let’s say you’re traveling 50 miles an hour, and point B is 500 miles away. The math is easy enough here that you know it’ll take you ten hours at that speed. But you could express this as a model if you were so inclined. The building block for your speed could be labeled “v” (for velocity), the building block for your distance traveled could be labeled “x” (because mathematicians like to use “x” in these cases), and the building block for time could be labeled “t”. The mathematical model for how far you travel (x) in a given amount of time (t) at a given speed (v) is then written as x = v times t. 50 miles per hour times 10 hours = 500 miles. That is a mathematical model.

While models don’t have to be complicated, the ones that aren’t complicated usually aren’t that useful. You were able to figure out the answer above without having to go through all that algebraic nonsense. But suppose you started at point A with the car in neutral, and then you had to accelerate up to your cruising speed. And then you had to account for friction of the tires on the road surface, which changes over the course of the 500 miles you’re traveling. As does the wind that might help or hinder your car. As will the curves in the road, and changes in elevation along the way. And of course the traffic, which confounds even the best of our navigation apps. Eventually, the real world gets complicated enough that building a mathematical model is well worth the time. Such is the case with Earth’s atmosphere, and that is why modern weather forecasting centers around mathematical models.

Attempts to model the way our atmosphere behaves date back to the early 20th century, but it wasn’t until the advent of computers around the mid-century mark that implementation of these models became even remotely practical. If it takes ten times as long to model the weather as it takes for the weather you’re modeling to happen, that’s not much use. But computers have become ever more powerful since their early days, and now we can predict the weather pretty accurately several days in advance. For most of us, our personal experience with weather forecasts always focuses on when somebody got something wrong. But if you think about it, they’re usually not wrong, at least not to a level that would have changed your plans. It rarely snows a foot when clear blue skies were forecasted. Hurricane tracks have become remarkably accurate in the days leading up to landfall. Scientists are tirelessly working on ways to improve our models of the weather. Some of it is just about buying more powerful computing systems. Some of it is about getting more accurate measurements from radars, satellites, and radiosondes (more about that below). And some of it is about developing better mathematical approximations of the physics. As it turns out, even if we had perfect measurements, perfect equations, and infinite compute resources, the forecast would still degrade the farther out you go in time. That’s because the things that happen at the molecular level are chaotic, and that chaos spreads over time like ripples in a pond (the “butterfly effect”).

A key to any mathematical model is how it is initialized – meaning how we state the conditions at the start of the modeling process. In the simple car example above, the initial conditions could have been stated as “the car is moving at a speed of 50 miles per hour when it leaves point A”. For weather models, we have to state things like “the temperature and humidity at various heights in the atmosphere were this”, and we’d typically get that information from radiosondes (ballon-borne instruments that take measurements as they go up) or satellites (which use photons of various types to infer the temperature at various heights). The more accurate you initialize a model, the better its results will be. Or more succinctly, garbage in, garbage out. As long as we continue to invest in satellite technology and other types of observations, we’ll continue to move farther and farther away from garbage in, and our forecasts will continue to improve, although the butterfly effect will keep them from ever reaching “perfect”.

Just like we model our weather (what’s it going to be like over the next few days), we also model our climate (what’s it going to be like over the next few decades). Asking “what will the globally averaged temperature be in twenty years” is very different from asking “what will the temperature be in Denver at 2pm this Saturday”. The dominant forces that determine the answer to the first question are different than the dominant forces that determine the answer to the second. And because it’s over a broader scale both in space and in time, the effects of chaos are dampened out in the question of climate. So even though an accurate weather forecast can only go out to a couple of weeks at best, a properly constructed climate model can go out to many decades. We know this because we’ve been able to validate the accuracy of our climate models by running them on the past and comparing them with what we know has happened. That gives us confidence to talk about the future, which means we can deduce from our current rate of climate change what the world might very well look like, say, by the year 2100. In the case of the studies carried out by the Intergovernmental Panel on Climate Change (IPCC), we’ve run a lot of models with a lot of different initializations, and with a lot of different assumptions about the rate at which we continue to inject (or not) more carbon dioxide into our atmosphere. Again, I urge you to take a look at all they’ve done on their website. I’ll only summarize it here, with pictures from the IPCC’s latest report.

First, what is climate change already doing to our world? The picture below is jam-packed with answers to that question, and no amount of words from me can add to that.

Next, where are we headed? That depends primarily on what we decide to do as a species. The leading scientists on this issue have modeled a range of such behaviors, each one with a different Representative Concentration Path (RCP) for carbon dioxide emissions. A specific working group (WGIII) was tasked by the IPCC with looking at a range of carbon dioxide concentrations as part of this process. Carbon dioxide is typically measured in parts per million. Today, it averages a little over 400. For reference, when I was in grad school running atmospheric models myself, a typical value was in the mid 300’s. The picture below summarizes the findings of the IPCC and WGIII for various scenarios, ranging from stringent reductions in greenhouse gas emissions (RCP2.6) to no efforts at all (somewhere between RCP6.0 and RCP8.5). In graph (a) below, the lines indicate these different paths, and the shaded areas indicate the resulting range of carbon dioxide concentrations. Graph (b) shows the resulting temperature change for these ranges of carbon dioxide concentrations.

Finally, the picture below summarizes what we’ll be dealing with if climate change continues at its current pace.

These are not just isolated models generating the results shown here. They are collections of models run over hundreds of scenarios – supermodels, if you will. The scary part? Things like the disappearance of ice in the Arctic are happening even faster than what the models had previously forecasted. So while it’s fair to argue that we don’t know for sure where climate change will take us, if anything we are being conservative. The next (and final) post in this series will talk about what we can do to change the current course, and why climate change is only one reason to do so. But for now, take a moment to listen to the supermodels.