Mathematics and Climate by Hans Kaper and Hans Engler SIAM Other Titles in Applied Mathematics Vol. 131 295 pp. (2013) ISBN: 978-1-611972-60-3
	Reviewed by Steve Schecter Department of Mathematics North Carolina State University

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This is an ambitious book. Its purpose is to “introduce students to mathematically interesting topics from climate science.” The “target audience” is “advanced undergraduate students and beginning graduate students in mathematics.” Climate science is a hot subject because of global warming, but the book's interest is broader.

I approached the book with the question, could I use it to teach an undergraduate course? I'll first try to give an idea what is in the book, then come back to this question.

The book has two intertwined tracks: introduce relevant areas of mathematics and statistics, and show how they are used in research papers and practice in climate science. The mathematical and statistical areas presented include:

• Qualitative theory of ODEs.
• Bifurcation theory.
• Linear regression, including analysis of residuals.
• Fourier analysis, including the fast Fourier transform.
• Spectral analysis using Legendre polynomials.
• Equations of hydrodynamics in the presence of the Coriolis effect, and shallow water approximations.
• Using spectral analysis to reduce a PDE to a system of ODEs.
• Delay differential equations.
• Advection-diffusion equations.
• Statistics of extreme events.
• Data assimilation, which requires an excursion into multivariate Bayesian statistics.

These topics are presented in chapters or parts of chapters that in some cases amount to mini beginning courses. Whew!

Payoffs from climate science include:

• Simple conceptual models that use small systems of ODEs to represent (1) the earth's energy budget, and (2) transfer of heat and salt between ocean basins (thermohaline circulation).
• The Lorenz system.
• Use of regression to understand the atmospheric carbon dioxide record from Mauna Loa Volcano on the island of Hawaii and to treat times series from changing sources, for example when a temperature gauge is moved.
• Milankovitch's theory of how changes in the earth's orbital parameters cause ice ages, which can now be tested using spectral analysis of reconstructed historical temperature data from ice cores and long-time numerical integration of a model of the solar system.
• A model for the Earth's temperature profile by latitude.
• Two models, using ODEs and delay differential equations respectively, for the El Nino-Southern Oscillation.
• Determination of the dependence of the fractal dimension of Arctic melt ponds on their area.
• A PDE model for algal blooms.
• Use of order statistics and related ideas to determine whether the recent spate of high-temperature years is meaningful or random.

Lots of climate science background is presented to prepare for these examples, and there are many exercises.

Could the book be used to teach an undergraduate course? Undergraduate electives are a sweet spot in the mathematics curriculum: it's okay to show interesting pieces of mathematics without thoroughly developing them, which is more of an obligation in a graduate course. Thus an undergraduate elective in mathematics of climate, in which nuggets of mathematics are presented and used to interesting effect, sounds like a great idea.

Could it work using this book? More precisely, could the required pieces of mathematics and science be explained in a reasonable amount of time, and would the resulting insights feel sufficiently powerful to be worth the effort? For an elective undergraduate course, the last part of the question is important. For example, the book includes a development of two-dimension dynamical systems with an application to thermohaline circulation. The result is the same bifurcation diagram that was earlier obtained from a model using just one differential equation, although with a different meaning for the parameter. I think undergraduates would find this an anticlimax.

So what material would work? I would want to include:

• One-dimensional ODE models for the earth's energy balance and thermohaline circulation. The energy balance model has three equilibria for a range of parameter values, two of them stable. One of the stable equilibria represents our present climate, the other is 50 degrees centigrade colder. There is geological evidence that the Earth has actually been in the “snowball” state several times in the past.
• Uses of regression in climate science.
• Fourier analysis and the Milankovitch cycles.
• Statistics of extreme events, since the question of whether recent extreme weather proves anything about climate change is in the news.

In addition, I think I would discuss the equations of hydrodynamics in the presence of in the presence of the Coriolis effect, shallow water approximations, various types of waves, and the use of these ideas to produce delay differential equation models for the El Nino-Southern Oscillation. This would be challenging but would give the feel of getting to the heart of climate science.

The text includes Matlab codes for the Lorenz equation, regression analysis, and delay differential equations. The last two would be a help in my proposed course. A website for the book at http://faculty.georgetown.edu/engler/mathclimate/textbook.html has errata and some additional exercises.

When one is thinking of offering a new course, a big help is information on the web about other similar courses. According to the website, the book has been used in courses at Arizona State and the University of Utah in addition to the authors' institution, Georgetown, but I could not find information about any of these courses on the web.

I would never have considered offering a course in mathematics of climate before this book appeared. The authors have done a Herculean job to make such a thing possible!