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This compact book is intended to introduce students in Earth Sciences to the procedures and techniques of mathematical modelling, that is, ‘…to teach postgraduate and advanced undergraduate students the skills necessary to represent complex Earth systems with mathematical and computational models that provide enhanced insight into processes and products’ (Preface). This it does in a simple, thorough and admirably systematic way, introducing a graduated sequence of methods and examples that both cause the reader to recall or refresh the year of calculus required for the mathematical treatments, and lead him or her through useful experiences of modelling. There is no doubt that the intended audience will find this volume an ideal introduction to the field, as a number of other reviews have emphasised.

A review such as this has two purposes; to advise prospective purchasers about the quality and value of the book, and to point out any shortcomings. The first done, I move to the second. Here, while in no way undervaluing the contribution this book can make to the technical education of its readers, I feel an opportunity has been missed to put the role of mathematical models in the Earth Sciences in a modern and wider perspective, by asking the question: ‘What is the value of mathematical modelling?’ This the book does not do – rather it focuses entirely on the techniques and procedures. This may be considered appropriate for a primer, but on the other hand, if the topic is not put into a more general context in a primer, where will it be so treated? More advanced texts are likely to assume that the question has already been dealt with. In my opinion students need to be made aware of the limitations and inherent traps of any technique during, or preceding, its introduction.

This book has the merit that it makes explicit the assumptions used in the various examples, which is the crucial start of the required contextualisation; but it fails to examine critically the way in which these assumptions may affect the degree to which the solution might depart from the real behaviour of the system being modelled. For example, in a model of a lahar, the assumption is made that the rheology of the lahar material ‘is similar to that of water’, but the nature of this similarity is not explored. It is later assumed that the flow of the lahar is turbulent; again, the implications of this are not explored. In reality, lahars may have distinctly non-Newtonian rheologies and if their flow is non-turbulent this may affect their behaviour substantially. This point is certainly within the grasp of the target audience, so why is it not dealt with? It would be invaluable for the reader to be led through a process of assessing whether the assumptions lead to over- or under-estimates of lahar depth, velocity and runout. If the objective of mathematical modelling is to provide ‘enhanced insight’, then a physically unrealistic basis is unlikely to do this; if the objective is to delineate a hazard zone, the consequences of lack of realism may be serious.

The modern concept of Earth systems behaviour is that of complex dissipative dynamical processes that are nonlinear; sensitive to initial conditions (chaotic); emergent (not predictable from subsystem behaviours); and better represented by numerical simulations than mathematical models. This view in no way devalues mathematical models and the contributions they can make, but it does accept that the equations that represent many Earth systems are insoluble mathematically and, thus, emphasises the limitations of mathematical modelling. Nonlinear systems receive rather perfunctory treatment in this text, while chaos and complexity are not mentioned; occasional references to mathematical models as simple, rapid ‘toy’ models that may provide useful insights are generally not accompanied by examination of the possibility of misrepresenting the system behaviour.

It seems to me that a great opportunity has therein been missed to set out for students a realistic appreciation of the intrinsic complexities of Earth system behaviour, and thus enable them both to choose when and when not to use mathematical modelling techniques, and to think about the likely accuracy of their predictions. Perhaps the authors will consider this in the second edition of what is, in spite of the shortcomings which I have probably overstated, likely to become a very popular and justifiably much-admired text.