Tuesday, November 4, 2014

Mathematics and Climate

Review of the new book Mathematics and Climate by H. Kaper and H. Engler:

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.

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