Mathematics of Planet Earth

Dr. Emily Shuckburgh, the leader of the Open Oceans research group in the British Antarctic Survey, gave a terrific talk on the mathematics of climate science here in San Diego on the opening day (January 9) of the Joint Mathematics Meetings of the American Mathematics Society and the Mathematical Association of America.

Emily’s talk was entitled “Using mathematics to better understand the Earth’s climate.” She was introduced by Christiane Rousseau, who gave us a great preview of the other MPE2013 activities still to come at the JMM. (You can see the full list here.)

One of Emily’s main points was that basic physics and simple mathematical models go a long way towards explaining the Earth’s overall surface temperature, but that more sophisticated concepts from dynamical systems and differential equations are important in modeling the finer features that we need to understand in order to get a picture of the changing climate. Her other main point, just as important, was that mathematics alone is not enough; you have to have a good understanding of the underlying physics of the Earth’s climate system to help you understand what is important to model.

She started with the 1827 work of Fourier, whose simplest model of the Sun-Earth system, ignoring the atmosphere and just using black body radiation and Stefan-Boltzmann, predicts an average temperature of 255 kelvin ($-18^{\,\mathrm{o}\,}$ C), whereas the actual observed temperature is about 288 kelvin ($15^{\,\mathrm{o}}$ C). (One kelvin (K) equals one degree Celsius (${}^{\,\mathrm{o}}$ C).) She then put in a simple model for the atmosphere’s transmission and absorption of radiation (ignoring convection) and showed that the prediction becomes 286 kelvin, which is remarkably close to the observations for such a simple model. Of course, two degrees on a planetary scale is still quite significant for us humans trying to live on Earth!

Emily went on to point out that the calculations were based on the known absorption/radiation characteristics of the atmosphere, which would change if the composition of the atmosphere changed, in particular, if the percentage of carbon dioxide in the atmosphere were to change. She showed us how sensitive the Earth’s climate is to this by giving us a short tour of what is known about the historical correlation between the carbon in the atmosphere and the Earth’s average temperature, a history based on ice cores that cover the past 800,000 years. This bore out the the model’s effectiveness and showed how robust the calculations are while, at the same time, pointed out how important it is to know the level of greenhouse gases in the atmosphere.

She pointed out that the correlation doesn’t say anything about causation, because, while increasing the carbon in the atmosphere would increase the predicted temperature, increasing the temperature would, according to our models, most likely increase the amount of carbon in the atmosphere.

To understand the dynamics, it’s very important to understand how heat is transported around on the Earth’s surface, particularly from the equator to the poles. (After all, there’s a lot of ice at the poles, but moving a lot of heat there is likely to cause dramatic melting.) So Emily then went on to describe the models of this transport, using basic Navier-Stokes models, first for the atmosphere and then for the oceans.

This is where her presentation became even more interesting. Even for the ‘simpler’ atmosphere problem, the rotation of the Earth causes instabilities to form in the solutions to the fluid convection equations, leading to such phenomena as the jet streams and the boundary instabilities that she described as ‘eddies’ (aka ‘swirly patterns’, roughly about 1000km in size) that can drive heat flux and help transport heat from the equator to the poles. She then described the eddies in the ocean system (which are much smaller, around 25km in size, but still very important for a good model); this is a much greater challenge, because numerically, you have to have a much, much finer grid to get the necessary resolution.

The dynamical systems that then show up, both in models and in observation become very interesting. Of course, no one can solve them explicitly, but known dynamical features such as KAM tori do show up and form barriers to mixing, so there’s the challenge of understanding the interaction of the regimes of strong and weak mixing. It was amazing and illuminating to see how these basic ideas from dynamical systems come into play in understanding the models and in motivating the scientific observations that need to be made in order to improve our models.

Emily went on to describe some of her own theoretical and observational work in developing and testing models of how the ocean currents, particularly around Antarctica, are distributing heat and what some of the likely effects will be on a warming planet.

She concluded with some sobering statistics of what might happen to our climate if the atmospheric carbon continues to rise, and pointed out that we are NOT on track to rein in the human contribution to atmospheric carbon, so we are potentially on the verge of climate regimes that we have not seen in human history, so there is a serious potential for rapid and dangerous climate change.

At the conclusion of this exciting talk, the audience showed its appreciation by giving her a rather long ovation, and we all left with a much better understanding of the mathematical and political/social challenges we face.

Robert L. Bryant, Director
Mathematical Sciences Research Institute
17 Gauss Way
Berkeley, CA 94720-5070
MSRI
bryant@msri.org