A Nobel laureate warned me, some time ago, to be on the alert for the stage at which “your mathematics departs the physics.” Because physics and chemistry underlie climate science, mathematics plays, to say the least, a prominent role in models and theories of climate.
Perhaps the first thing to appreciate about this book is the authors’ clear statement of purpose: “to introduce students to mathematically interesting topics from climate science.” Interesting is an understatement; in fact, the mathematics presented here does not depart the science in my teacher’s sense, that is, over and above the inevitable and inherent limitations of theories and models comprising the current state of the art. (Derman [1,2] treats theories and models, and the distinction with a difference between them, in a masterly fashion. I’ll not dwell on that distinction, as the underlying theories here comprise established bodies of knowledge, even as the models are the mathematical and conceptual models well elucidated in the book.)
The second thing to appreciate is the non-dilution of scope: this is not a general climate science book, but rather an excellent and focused treatment of mathematical topics that continue to arise naturally from climate research. A third attractive feature is the strong deemphasis of normative judgments regarding global warming or climate change. The authors note that “decision makers have more questions than science can answer.”
The 20 chapters will certainly provide insight for climate scientists, but will undoubtedly also enhance the skills and knowledge of students and practitioners of computing, mathematics, physics, chemistry, and branch fields. The chapters include: 1) “Climate and Mathematics,” 2) “Earth’s Energy Budget,” 3) “Oceans and Climate,” 4) “Dynamical Systems,” 5) “Bifurcation Theory,” 6) “Stommel’s Box Model,” 7) “Lorenz Equations,” 8) “Climate and Statistics,” 9) “Regression Analysis,” 10) “Mauna Loa CO2 Data,” 11) “Fourier Transforms,” 12) “Zonal Energy Budget,” 13) “Atmosphere and Climate,” 14) “Hydrodynamics,” 15) “Climate Models,” 16) “El Niño-Southern Oscillation,” 17) “Cryosphere and Climate,” 18) “Biogeochemistry,” 19) “Extreme Events,” and 20) “Data Assimilation.” Appendix C has some short MATLAB code examples, of which I found the code for delay differential equations most interesting.
Though computers have made finding numerical solutions to complex equations evermore practicable, the authors here take a systems-level approach to taming the ultra complexity of climate. Here, “mathematics can offer perspectives that ... complement or provide insight into the results of observations and large-scale computational experiments.” The discussion of the overall energy budget of the earth, the global energy balance model, will hold anyone’s interest, as it is vivid and clear. The physics of the greenhouse effect and similar phenomena is well explained, and all equations are well presented. Bifurcation and multiple equilibria are consequences rendered palpable by the well-organized exposition. The exercises throughout the book are real in the sense that they involve real science such as Planck’s radiation law and the Navier-Stokes equation.
The treatment of the oceans’ influence is equally satisfying, as it works around the absence of a universal equation of state among the water’s density, temperature, and salinity. The notion of a box model is introduced here, as well as an instance of a dynamical system. The ensuing treatments of dynamical systems and bifurcation theory are ones that I would have preferred as my first exposure to these subjects; together with the exercises, they stand very well on their own as mini textbooks. The same can be said for the brief chapter on the 1960s Lorenz equations, which I recall vividly as putting (the mathematics of) chaos on the map, and on television.
Statistics is, predictably, another large topic in the book, for which climate science poses “uncommon challenges” that are both “methodological and practical.” For example, randomized comparative experiments are impossible here. The formulation and testing of clean hypotheses is also difficult to impossible, as is the positing of probability models. The book thus offers numerous special techniques.
The treatment of (fast) Fourier transforms is, again, the epitome of clarity and efficiency, qualifying this section as another mini textbook. The application to glacial cycles serves well to reinforce the Fourier techniques.
It is difficult to select a favorite chapter, but my first thought would be chapter 14, on hydrodynamics, perhaps because that subject left me with a residual fear that failed to give way to my desultory exposure to it. This perfectly paced, pithy chapter will serve to welcome many of us nonspecialists back to this generally neglected topic, which, though classical, has unsolved problems such as Navier-Stokes at its frontier.
Benjamin Franklin once exhorted us to imitate Socrates. In my version, as it applies to science, it would be Feynman, who always kept a dozen techniques in his back pocket for ingeniously supplying a solution to a new problem. I highly recommend this book in general, and also as a source of techniques for your back pocket.
