The Mathematics of Planet Earth

great wave

Rogue waves in art and nature. Image: J.M. Dudley, V. Sarano, F. Dias.

Happy Earth Day! It seems appropriate today to highlight the Mathematics of Planet Earth blog. In fact, it’s triply appropriate: today is Earth Day, April is Mathematics Awareness Month (with a theme this year of the mathematics of sustainability), and 2013 is the year for Mathematics of Planet Earth.

MPE is an international initiative to promote 2013 as a year for research into the mathematics underpinning everything from climate change to disease patterns to celestial mechanics. The initiative has a strong outreach component: planetary problems can introduce to the public and motivate general interest in mathematics, and public presentations about the mathematics of climate change and other such problems may reduce polarization around these topics. (Or, as the comments on a recent article I wrote about MPE indicate, maybe they won’t. But perhaps least such presentations can educate those with an open mind.)

Like the initiative topics, the MPE blog posts are diverse. Many give updates on current events, and some are expositions about recent research or general overviews of MPE topics. I especially appreciate that the expository pieces are clearly written for a mathematical audience and don’t shy away from showing us some of the equations under the hood! The blog is updated almost every day, so there is a lot to explore.

Hans Kaper, an applied mathematician and professor emeritus at Argonne National Laboratory, is the designated blog wrangler. He is still accepting submissions of new posts. With more than 250 days left this year, there’s plenty of time for you to write a post and plenty of room on the blog for your contribution. Personally, I’d love to see more about the history of all the MPE topics and more expository pieces about mathematical biology and ecology. Christiane Rousseau is the catalyst of the MPE initiative, and her blog post What is an MPE Topic? might give you some ideas as well.

In chronological order, here are five MPE blog posts I’ve particularly enjoyed.

Prospects for a Green Mathematics (also covered by my co-blogger Brie Finegold in the February Math Digest)

John Baez and David Tanzer eloquently describe the climate change and sustainability problems facing our planet and species. “Where does mathematics fit into all this? While the problems we face have deep roots, major transformations in society have always caused and been helped along by revolutions in mathematics.” They go on to propose the term green mathematics for “mathematics suitable for understanding the biosphere” and, focusing on network theory, disabuse us of the notion that green mathematics is actually just biology or ecology, with another name.

To illustrate, Baez and Tanzer include two concrete examples with helpful pictures. I particularly enjoyed the optimal transport research of Qinglan Xia (full disclosure: Xia and I both went to Rice, although we graduated 10 years apart). “Unlike approaches that merely create pretty images resembling leaves, Xia presents an algorithmic model, simplified yet illuminating, of how leaves actually develop. It is a network-theoretic approach to a biological subject, and it is mathematics—replete with lemmas, theorems and algorithms—from start to finish.” The coolest thing is that tweaking parameters in the models Xia uses leads to pictures that bear a striking resemblance to real leaf species.

There Will Always Be a Gulf Stream—An Exercise in Singular Perturbation Technique

Robert Miller of Oregon State University walks us through the nuts and bolts of a dynamical model of the Gulf Stream. “The Gulf Stream and its analogs in other ocean basins exist for fundamental physical reasons. Climate change may well bring changes in the Gulf Stream. It may not be in the same place, may not be of the same strength or have the same temperature and salinity characteristics, but as long as the continents bound the great ocean basins, the sun shines, the earth turns toward the east and the wind blows in response, there will be a Gulf Stream.”

The Great Wave Explained by Directional Focusing

“One of the most famous images in Japanese art is the Great Wave off Kanagawa, a woodblock print by the Japanese artist Hokusai. The print shows an enormous wave on the point of breaking over boats that are being sculled against the wave’s travel (see Figure 1a). As well as its fame in art, this print is also famous in mathematics: firstly because the structure of the breaking wave at its crest illustrates features of self-similarity, and secondly because the large amplitude of the wave has led it to be interpreted as a rogue wave generated from nonlinear wave effects (see J. H. E. Cartwright, H. Nakamura (2009) Notes Rec. R. Soc. 63, 119–135).”

John M. Dudley and Frédéric Dias discuss their recent paper challenging this idea that rogue waves have to be generated by nonlinear effects, describing directional focusing as an alternative linear explanation. As a bonus, they collaborated with photographer V. Sarano to get a picture of a wave that looks a lot like the wave in Hokusai’s print, and as you can see at the top of the post, it’s magnificent!

Chaos in an Atmosphere Hanging on a Wall

Christopher M. Danforth of the University of Vermont marks the 50th anniversary of Ed Lorenz’s paper rigorously proving the unpredictability of the weather. “While many scientists had observed and characterized nonlinear behavior before, Lorenz was the first to simulate this remarkable phenomenon in a simple set of differential equations using a computer. He went on to demonstrate the limit of predictability of the atmosphere to be roughly two weeks, the time it takes for two virtually indistinguishable weather patterns to become completely different. No matter how accurate our satellite measurements get, no matter how fast our computers become, we will never be able to predict the likelihood of a rainy day beyond 14 days. This phenomenon became known as the butterfly effect, popularized in James Gleick’s book ‘Chaos.‘”

Danforth includes two videos from his computational fluid dynamics lab at the University of Vermont, demonstrating how chaotic behavior arises in a rather simple experiment.

Why is celestial mechanics part of MPE2013?

I think the title of this post by Christiane Rousseau undersells it a bit. Beyond a simple answer to that question (Earth is a planet, and it moves in space), Rousseau gives a great crash course in the n-body problem, KAM theory, and the current understanding of the instability of the inner planets.

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