Mathematical Geoscience

The first thing to know is that this is not just a book about rocks. Indeed, there are hardly any rocks at all. It’s true that the title doesn’t offer much of a clue about good stuff inside. The author confesses that he struggled with the title, and it’s not too hard to see why. His working title was “Mathematics and the Environment”, and although that sounds fashionable, it does not do justice to the breadth of the effort. The book is a comprehensive overview of the application of mathematical models to problems in the environment, and “environment” includes everything from the top of the atmosphere to deep in the earth — and more. All the models that the author considers are based on differential equations.

The scope of the book is truly amazing: topics range from climate modeling to magma transport, landscape evolution, the ocean, the atmosphere and a great deal more. There are even a few places — talking about the north polar cap or the dunes on Mars — where we go beyond the “geo” part of geoscience. There is so much material in the book that many of the book’s chapters could be developed into entire volumes of their own. Yet the author treats his material succinctly but not superficially.

A typical chapter begins with a short description of the physical characteristics of an environmental feature and then proceeds to describe and develop the critical elements required to model it. For example, the chapter on river flow begins with a discussion of the hydrological cycle and the issue of turbulent flow, and proceeds to Chézy’s and Manning’s laws that describe stream velocities and the St. Venant equations that represent conservation of mass and momentum. The author then considers linear river waves, and finally turns to consider unusual nonlinear waves in rivers such a roll waves and tidal bores.

A broad theme that permeates all the applications is transport — of energy, water, air, sediment, magma, glaciers. There are too many applications even to summarize adequately, but I found a few that caught my fancy. The author does a particularly nice job of capturing the important ideas of global climate modeling in just under sixty pages, and the result is well worth reading. Not much of it is available outside the more specialized literature.

I also learned a good deal about dunes, the desert variety as well as the kind that occur in riverbeds. The tidal bore in the river Severn in England is fascinating — something that has intrigued me ever since I saw a smaller version near Blue Hill in Maine. Finally, I’ve found that I always learn great new words when I dip into earth science. The new one I learned here is “jőkuhlaup”, an Icelandic word for a catastrophic outburst flood from a glacier.

This book is a lot of fun, even just to browse. It also has a wealth of examples of applications of ordinary and partial differential equations. The book is addressed to readers having some familiarity with differential equations and advanced calculus. Basic classical physics and a bit of chemistry would be useful background, but no special knowledge of earth science is assumed. The writing is at least a couple of notches above the norm for a textbook. The exercises are another strong feature of the book: they are creative, numerous and challenging.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.