Nonlinearity and the Coming Deluge

delugeThe image above shows part of Minot Air Force Base, from a story about how they are recovering from a flood. This week’s post is about a problem with the way we write and interpret models of climate change. We may be in far more trouble than we think because of non-linear effects. A system is linear if the change in the output of the system is a simple multiple of the change of the input of the system. Non-linear systems are those that are far from being linear. Here is the source of our problems with non-linear systems: they often look linear over short periods of time.

We have a problem here.

As the temperature rises, things melt, plants have longer growing seasons, and natural systems pass critical thresholds. Think about ice melting. You start with a solid. You warm it and, for a while, it simply becomes a warmer solid, obeying a simple linear relationship of heat input to material temperature. At some point, however, the ice turns into water, shorts out the heater you are using, and starts an electrical fire on the lab bench. At least, this is what happens if you don’t understand the behavior of ice yet — which is the position the human race is in with respect to climate change.

There is another fact that may seem a bit radical: reducing our carbon footprints is pretty much worthless to prevent large-scale destructive climate change. This is an argument Occupy Math has had with innumerate environmental activists several times. Reducing our carbon footprint would not hurt, but the genie is out of the bottle. There are many good reasons to reduce your carbon footprint — but preventing climate change in the next five decades is not one of them.

Let’s do the math!


Look at the graphs above. The left graph is an input:output graph where increasing the input increases the output. Notice that the increase in output is smaller at higher levels of input. This is a simple type of non-linear system. If we zoom in on one a little piece of the curve, the graph on the right, it looks a lot like a line. At the scale shown in the zoomed in graph, a line is a good model of what is going on. This effect of looking linear on a local scale is pervasive and, used correctly, it is a valuable mathematical tool that makes some very hard calculations possible to do.

So where is the bad?



Since carbon dioxide traps heat, the short-term effect of more CO2 in the atmosphere is higher average temperatures with a simple linear relationship. You can measure the effects over time of the CO2 on the temperature, fit a line, and get a model that lets you estimate what sort of reductions in carbon dioxide emission are needed to achieve goals for controlling yearly average temperature. The problem is that this is probably an incorrect model. The point in the graph that says “Something?” exists — in fact several possible “Somethings” exist.

Rotting Arctic permafrost.

The Arctic has a lot of places that are permanently frozen, but which used to be composed of living plant matter. As the world warms up, more and more of the permafrost melts. The thawed plant matter rots, emitting methane. Methane is better than carbon dioxide at trapping heat, about 25 times more effective. This is a positive feedback loop. Higher temperatures release more methane which raises the temperature even more. The effect does end, when most of the permafrost has finished rotting, but the process ends at a higher temperature. The rotting of the arctic permafrost is a big potential something.


The Clathrate gun hypothesis.

Methane clathrates are solid ice-like complexes of water and methane on the ocean floor. The picture above shows marine worms that live in methane clathrates. Their intestinal bacteria digest methane for the worms. These deposits sequester 530,000,000,000,000 tons of carbon — that we know about. These deposits can be destabilized by warming. A controversial hypothesis posits that catastrophic release of methane from the clathrate deposits is possible within a single human lifetime. Release of carbon from clathrates is thought to be slower than the Arctic permafrost release, but the total amount of carbon is much larger. This is an area in critical need of more study.


Ice and Earth’s albedo.

All the heat we are worrying about comes from the sun. In the distant past, earth was covered with ice which reflected away the sun’s light, keeping the temperatures low. Climate change is melting the polar ice caps, especially the one at the north pole. This makes earth better at absorbing the sun’s light. This is another positive feedback effect, and one that is really hard to measure accurately. In broad brush strokes this cannot help and probably hurts. It is also clear that the Arctic ecology is changing in big ways.

Enough with the horror stories!

What can we do about all this? The good news is that there are some pretty clear targets. Occupy Math sees three critical priorities.

  1. Learn to build reliable sea walls around coastal cities.
  2. Work really hard on food security.
  3. We need a global climate refugee policy.

The first priority is obvious. The second is one of the bright spots in all this. High efficiency grow lights made with new solid state technology mean that we can build green-houses the likes of which have not been seen before. This is evidence the food problem is solvable. We also need a whole lot of good old county extension type work to help farmers have a good idea of what crops will grow at their location next year. The correct choice of crop in a given location is starting to change faster and faster as the climate changes.


The third priority is being broadly ignored. The horrible Syrian civil war is arguably the first modern climate change war. While there are many political components to the Syrian civil war, it started after a series of crop failures. The country of Iran is having huge climate-driven agricultural problems. The list of crop failures goes on and on. Wars started by reactions to climate change driven starvation could kill billions of people over the next century. The chaos will be highest near the equator and the worst in poor countries. We have a good chance, on our current trajectory, of having a planet with large areas that are uninhabitable outside of huge, expensive controlled environments. A good refugee policy could reduce the tragedy substantially.

Do something!

For most people, becoming woke on these issues is the best first step. Help others learn about the coming deluge of both sea water and refugees. Do your best to explain non-linearity and its implication that it is very late in the day for climate change. There are a bunch of things we can do and some of them are pretty far out. Learn to like eating insects, for example. They can be raised in greenhouses, they are better for you than most other types of meat, and they have much lower environmental impact. There are carbon capture technologies that may help. Just now they are way too small, but we are an innovative species.

Occupy Math hopes this has been educational. Do you have other examples of “Somethings?” Do you want to hear more about positive actions you can take? Do you disagree that the genie is out of the bottle? Please tell us about it in the comments.

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics


One thought on “Nonlinearity and the Coming Deluge

  1. “In broad brush strokes this cannot help and probably hurts.” Actually, the melting of the Arctic Ice Cap can help. Note that at latitudes like for the Arctic Ocean, the sun’s angle of incidence is low and ocean albedo is higher a low angles of incidence, reducing the positive feedback from the extra absorption of solar energy. Consider also that ocean currents transport heat poleward and that the ice cap was insulating the ocean. Open ocean allows more heat to be radiated into space, especially as it extends longer into the autumn months. Furthermore, this paper found that cloud cover over the open ocean was higher in the spring, canceling some of the positive albedo feedback. See figures 2a and 2b, detailing the short and longwave radiative fluxes. When the open ocean in October and November is factored in, the net feedback from the Arctic ice cap melting may be close to neutral or even negative.


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