All posts by Goodwin Gibbins

I'm doing my MRes and PhD on the mathematics of climate change at the moment, trying to step back and look at the system as a whole. Current main areas of interest are paleoclimate, thermodynamics of the earth system in general, entropy in particular and what scientists can do to help society best with this global warming problem.

FAQ: Why do HFCs and CFCs contribute more to warming?

This question came from a young relative of mine doing a school project on climate change and it’s a really interesting one, bringing in a lot of important concepts about the climate that aren’t explained as much as they should be. Here we go!

Ultimately, it’s the sun that warms the planet. Well, okay, there’s some heat coming from the core, but the crust’s rock layer does quite a good job insulating so we can pretty much neglect it. I think we can agree that the sun is the hot thing – much much hotter than the ground.

As the sun heats up Earth, Earth needs to be being cooled down in some other way, otherwise it’d just get indefinitely warmer. There’s enough heat coming in from the sun that if we didn’t lose any, the surface of the planet would get considerably hotter every day!

We lose it through “earth-shine” or outgoing long-wave radiation, in the proper lingo. In the same way that the sun gives off light and heat we can feel when we go outside, the earth is giving off its own electromagnetic waves, just at a wavelength we can’t see. Wavelength is like colour: red, blue and yellow light are all different colours, but so are x-rays and radiowaves and what wifi and cell phone signals are transmitted through.

https://marine.rutgers.edu/cool/education/class/josh/em_spec.html

But since the sun-shine and earth-shine are at different wavelengths, they’re blocked by different things. Glass isn’t see-through to all light and nor is the atmosphere. Let’s take a close look at the diagram below:

http://cybele.bu.edu/courses/gg312fall01/chap01/figures/

The wavelength is measured in micrometers, which are million times smaller than a meter. The middle of the sun’s shine (“black-body”) curve is in visible radiation, very little of which is absorbed by the atmosphere (see the same region on the lower plot). For the earth-shine, it’s not so lucky. About 80% of the outgoing earth-shine doesn’t make it out – it gets trapped and re-emitted by the molecules in the atmosphere.

Those molecules work to block radiation because of their specific shape and how well they resonate (match) with the light at each wavelength. Water is really really good at it absorbing radiation at loads of wavelengths, but CH4 (methane) and CO2 (carbon dioxide) do a reasonable job too. They all block the outgoing radiation and make the surface of the earth heat up, like the glass in a greenhouse lets sunlight in but keeps heat from going out. That’s why we call them greenhouse gases.

Water has a special place because there’s so much of it coming in and out of the atmosphere in clouds and rain. We think of it as a response rather than a cause because you can’t inject it into the atmosphere and have it stay. The same is absolutely not true for carbon dioxide, methane and HFCs (hydro-flouro-carbons) and CFCs (chloro-flouro-carbons). They get added to the atmosphere by some natural processes and, unfortunately, by humans who have use them for fire extinguishers and refrigerants, and once they’re there, they stick around, blocking the earth-shine and changing the natural balance.

The question of how bad a particular greenhouse gas is is a difficult one. The first is whether it’s doing a job nothing else can do. The atmosphere has a lot of carbon dioxide and water in it, so adding a little more doesn’t make as much difference as adding something which absorbs in the gaps. If you look at the diagram below, you can see that CFCs (and the same is true for HFCs) absorb in the ‘atmospheric window’.

https://www.sciencedirect.com/science/article/pii/S0007091217334049

This means that every molecule added absorbs some radiation that would otherwise have gone through. Methane is powerful like that too. For CO2, a molecule added doesn’t have so much power – there’s already a lot of them so that each additional molecule doesn’t go so far. This is measured in the radiative efficiency.

The other thing that makes a molecule strong is how long it sticks around for (the lifetime). Methane turns into CO2 after a little while (10 years) in the atmosphere thanks to the active chemistry (driven in part by the sunshine) that goes on up there. Some CFCs and HFCs stick around much longer – check out the IPCC table here.

Now, suppose you put different things in the atmosphere and wanted to know how bad they are. You’ve got to combine two things: one is how much of a difference they make themselves (that is, how good they are at blocking earth-shine and whether anything else would have been blocking it anyway) and the other is how long it sticks about. That’s why we think about global warming potential.

From the EPA glossary

Global warming potential (GWP): A measure of how much heat a substance can trap in the atmosphere. GWP can be used to compare the effects of different greenhouse gases. For example, methane has a GWP of 21, which means over a period of 100 years, 1 pound of methane will trap 21 times more heat than 1 pound of carbon dioxide (which has a GWP of 1).

It adds up how much damage each gas does times how much of it is around over the course of 100 years. Take those together and you get a table like the one here. CFCs and HFCs come out pretty badly, with a GWP in the thousands!

But there is good news: we don’t make as much of them as some of the less nasty things, especially since the Montreal Protocol which came in in 1989. That’s one big success for global political agreements to curb climate change!

 

A puzzle: 100 year timeline over which to calculate global warming potential doesn’t do such a good job of taking into account your great-grandchildren! Over the molecule’s lifetime, something long-lived (like CO2) might do a lot more harm than something short-lived (like methane). So what should we prioritise?  Comparing 100-year global warming potentials or calculating warming per molecule over the molecule’s entire lifetime?

 

Why studying random dynamical systems matters

If I said I was studying multistability in random dynamical systems driven by Brownian motion, why should you care?

Martingales, filtrations, ergodicity, synchronisation, Monte Carlo methods: mathematics is full of words which invoke colorful ephemeral imagery for heavy, abstract concepts. Even if we explained the concepts and brought them to life, it wouldn’t be obvious how they were related to your world. Mathematics is powerful exactly because it exists a few steps away from reality, in a place where anything which obeys its structured rules is possible. But this means that to answer the question of why we study what we study (apart from that it’s beautiful and fun) we have to step out of mathematics and back firmly into the physical.

When we think about the Planet Earth mathematically we often think about understanding and capturing each of its parts, studying technical topics like turbulence, boundary currents, atmospheric waves, sub-gridscale processes or atmospheric chemistry. These are the building blocks of the Earth and investigating them shows us patterns and connections which let us project their behaviour forward a few days into the future by running them artificially faster inside our super-computers, predicting the weather or the dissipation of volcanic ash.

There is another powerful way to think about our Earth, though. For the past decade or so, biologists, for example, have stepped away from the reductionist  approach of looking at the components of their system (protein and DNA) to start looking at the high-level systems behavior. The pieces slot together intricately into such a complex system that some of its behavior (consciousness, for an extreme example, or an ant colony) is nearly impossible to discern by looking at the parts. They find that sometimes it’s better to drop the details and start at the top and work down, characterising as much of the emergent behaviour as possible since it’s part of understanding the animal, even if we can’t quite see where it comes from.

In the study of Planet Earth we’re lucky that we’ve been so driven early on by prediction even while we’ve been studying the pieces. We’ve always had an eye on the whole system, trying to simulate it in weather models since predictability is equivalent to safety and success as we try to flourish as a species. These models do an amazing job of mirroring the massive weather beast, especially as we tweak them year on year, checking them against reality and improving them. But what we don’t often do is explore theoretically the range of behaviour the complicated highly non-linear and multi-dimensional system we live within is capable of. We don’t often look at it from the top without the baggage of the details of the pieces. This is the truly “systems” approach.

Perhaps the simplest example of what you might see if you took a systems perspective is multistability: the idea that the Earth might have different states (or rhythms) it could comfortably fall into which would self-perpetuate themselves, even if in the same conditions another state would have been possible. The mathematical example is always a ball rolling down into a valley with another valley next door; the ball could stop at the bottom of either valley, both are stable states, but it can’t get from one to another easily. Ecosystems exhibit multi-stability – the same patch of land could be a birch forest or a grazed field and stay that way indefinitely. Even bodies exhibit multistability if you look at them from the right perspective: something alive generally keeps itself alive, but dead also stays dead.

A schematic view of bistability, in which the “ball” can be in either “valley” happily, but finds it difficult to transition between the two. Source: Wikimedia Commons.

Whether or not our climate has multiple stable states is vitally important because if it does and we start to push it, we could very possibly push it out of one and into another and it might be very hard to return. How would we know if it did? One way (the reductionist way) might be to carefully build a replica of it, to the very best of our ability, then start pushing it around and see if it switches. Not a bad idea, but with obvious drawbacks too: our model might not be quite perfect and it takes a long time to run these pushing experiments on a computer, so it’d be hard to draw solid conclusions. Another way is to think theoretically about what is required of a system to be a jumping system then study whether the Earth satisfies the conditions, looking back at the record we have about our real Earth system to see if we can see signatures of jumping. To do that, however, we need to know what jumping looks like.

That’s why we’re interested in a class of mathematical constructs called random dynamical systems. They’re an abstract construct and could be interpreted into the real world in lots of different ways so, like most everything in mathematics, they’re probably useful in a lot of fields. What they can do for us, though, is to represent the Earth like a point moving between states according to some update rule, complicated or simple, which summarises the evolution of our climate in time. What makes them even more complicated than the dynamical systems people like Ed Lorenz have been studying for decades is that we now allow the update rule to be a little bit random thanks to a Brownian motion term: to send the Earth up the valley when it should have gone down, occasionally. Suddenly, a whole new class of behaviour is exposed: with bistabilitity in the update scheme and additional noise comes stochastic resonance, the seemingly regular jumping of our proverbial ball between valleys depending on which way the wind (which wouldn’t usually be enough to move a ball between valleys!) blows. They also might simplify dynamical systems, with noise-induced synchronisation meaning that even if we don’t know where exactly the point started, it’ll end up in a similar place if it undergoes the same noise.

It’s not that the Earth is a random dynamical system (it’s a planet, not an abstract mathematical concept), it’s that we hope that by trying to represent it as one, we’ll learn more about it. Climate certainly has some random elements: weather provides unpredictable and sometimes large change in local climate, not to mention volcanoes. We also have stochastic resonance-looking jump-like events recorded in the history: the Glacial-Interglacials and maybe Dansgaard-Oeschger events in the past million years. If we find that to some extent our reality fits well into this concept, we will be able to draw conclusions from the concept back out to the real world. It probably won’t be as revealing as complex wave functions were for quantum mechanics, but having a new way to account for the world around us is almost surely a good thing.

When it comes to complex systems that we count on for our survival, no one approach can be relied on. We don’t have a reason to believe that there’s a single bit of mathematics that nature obeys on this scale, but there will be patterns and properties at all scales which fit into mathematical frameworks. Random dynamical systems let us look at the pattern of stability and large changes in the whole system and is, as both a mathematical field and a climate perspective, relatively new and so very exciting.

Why bring maths into it?

To thrive on planet Earth, knowing what it’s going to throw at us is key. We need to know what crops can be grown where and what time of year to plant and harvest them. We need to know whether the mosquito that’s about to bite us is likely to be carrying malaria. We need to know how to manage flooding, and how much snow or wind our structures have to withstand.

So how do we know these things? Experience, first. The security that what will happen tomorrow will probably not be too much different from things that have happened before. But more and more, we are profiting from a delicate and precise understanding of the “why” and “how” of the system encoded in mathematical models to deduce more exactly what tomorrow will bring: how much light will shine on the solar panels or even when to shut the roof on Wimbledon to avoid the rain.

With climate change, the need for strong predictive structures become even more dire. Humanity is taking a complex, intricate system and dramatically altering a key component – greenhouse gases. We really are putting a cat amongst the pigeons, and, without doubt, things are going to change. The future is not going to be like the past: how it will change and how we can best avoid the worst–those are questions that require a mathematical and physical mastery of the system.

And of course, this is desperately important. While we in Britain might have enough resources and padding to quickly adapt to these changes, a lot of the people who share the earth with us are in a much more precarious position. Think of what famine does to a country with ethnic tensions, what water shortages in a country with a strong military imply for its neighbors and how a poor country deals with a new set of diseases traveling in with the weather.

Realizing the patterns, quantifying the interactions and building models is not a fix by itself. We still have the problem of wanting ever more growth and energy at as low a price as possible, of prioritizing today over generations from now.

But what we, as mathematical and physical thinkers, can contribute is a demystification: revealing the behavior of the complex Planet Earth so that as a society, a species, we can make the large ethical decisions facing us with more determination and confidence.

Mathematics, physical, geological, biological, ecological and chemical sciences don’t offer a pre-packaged answers: they’re ways of thinking to be drawn on depending on the question at hand. In this rest of this blog, we’ll be talking about some of the puzzles important in a responsible response to climate change and what tricks and elegant techniques from the mathematical world especially we can use. Expect calculus in the form of Partial Differential Equations used to describe a system changing over time, linear algebra to make those equations approximately solvable by a computer, dynamical systems thinking to try to simplify the complicated evolution and understand the butterfly effect and finally statistics and probability to express what we know and don’t know after all of that.