Book reviews 921

dix. Full references to valuable papers and books over a wide range of subjects are given in 27 pages.

This is a useful book; however, I consider that neither the author nor the publishers have given it the optimum title.

M. .I. Rycroft International Space University

The Chemistry and Deposition of Nitrogen Species in the Troposphere, A. T. Cocks (Ed), 1993, x + 134 pp. Royal Society of Chemistry, f35.00 hb, ISBN O-85186-355-8.

This slim volume contains seven camera-ready papers pre- sented at a one-day Symposium organised by the Royal Society of Chemistry in February 1992. It is especially impor- tant and useful since nitrogen species play a central role in tropospheric chemistry. Fossil fuel combustion leads to photochemical oxidants (particularly ozone) and to photo- chemical smog. Nitrogen oxides and ammonia, mainly aris- ing from agricultural practices, lead to nitrogen deposition onto terrestrial ecosystems and adverse effects.

Current knowledge, a.nd its uncertainties, are well covered. Emissions from power plants generating electricity cause plumes downwind which have been both measured and mod- elled. Similar studies have been performed in central London, and at urban and rural sites in relation to emissions from motor vehicles.

The critical load concept is considered. Here, the sensitivity of the environment to pollutant loadings is derived and used to work back to estimate the amount of pollutants that can be emitted in the long term without causing environmental damage.

The behaviour of p,articulates containing nitrogen, and their deposition, primarily wet rather than dry, is considered. The final paper, on the results of modelling nitrogen com- pound concentrations lover Europe, and a comparison with observations, is particularly valuable.

M. J. Rycroft International Space University

Wave Packets and their Bifurcations in Geophysical Fluid Dynamics, Huijin Yang, 247 pp, 1991. Springer, New York f31.00.

The primary purpose of this monograph is presumably to present the author’s studies of wave-packet behaviour in the context of large-scale atmospheric or oceanic motion described by the quasigeostrophic equations. However, the account of these studies is preceded by an introduction to those aspects of the basic theory of wave propagation and of dynamical systems that are necessary to establish the author’s main results. In assessing this book, it is necessary, then, to consider both the plausibility of the central theme and whether or not the presentation of the introductory material is successful. The book has been typeset, presumably by the author himself, using LaTeX. It is perfectly legible, but there are a large number of minor typographical errors.

In Chapter 1 the author gives a very brief introduction to the equations of motion and the common approximations that arise in considering geophysical flows. A reader who was starting from scmtch with no previous knowledge of geophysical fluid dyna.mics would almost certainly need to use other more basic textbooks that treat this material in much more detail. The substance of the book really begins with Chapter 2, which gives an introduction to the theory of

waves on a slowly-varying basic state. This is quite satis- factory, though the treatment seems a little laboured at times. Some of this chapter, e.g. that concerning wave-action con- servation, is more a summary of the important results, rather than a step-by-step derivation and a reader who wished to verify the results for themself would probably have to consult one of the original references. The author chooses to give a detailed account of the Generalised Lagrangian Mean theory of Andrews and McIntyre. Although this is clear enough the detail seems misplaced in this case. After all, none of these results is used in the latter part of the book. It might have been better to make the general point that it is possible to derive wave-activity conservation theorems for finite-ampli- tude waves and leave it at that. It also seems fair criticism that the route chosen to deriving the wave action con- servation results is not the most convenient to apply to Rossby waves which are the subject of the later part of the book. It is surely most straightforward to proceed straight from the quasi-geostrophic equations.

Chapter 3 moves on to consider Rossby wave packets on barotropic basic states, starting with the shallow water equations. Various integral theorems for slowly varying wave packets are noted, though such theorems have analogues which are satisfied for general disturbances. The author introduces a definition of stability which is not standard and may cause serious confusion, particularly with regard to his Theorem I. (This definition allows him to pose the question ‘Is it necessary that a barotropically unstable wave packet always remains unstable?.) However the most substantial part of the chapter deals with the behaviour of wave packets in the presence of p (representing the planetary vorticity gradient), S (representing the variation in the planetary vor- ticity gradient) and topography. The author concentrates here, and later in the book, on the case where the spatial gradient of the dispersion relation is itself spatially inde- pendent. It follows that the differential equations predicting the time variation of the wavenumber components as a wave packet moves along a ray path have no explicit dependence on the position of the packet. This leads to some analytical simplification, but not, unfortunately, to notational sim- plification There is a tendency for the author to define large numbers of new constants and to repeat equations unnecess- arily, all meaning that it is difficult to jump into a particular chapter and make sense of the results, without having plodded through previous chapters in detail first. The qualit- ative behaviour described in the latter part of Chapter 3 is familiar in many contexts of wave propagation on shear flows and it is difficult to see anything very new here, apart from an exhaustive exploration of parameter space.

The wave packet theme is developed in Chapter 4, where it is noted that some basic states lead to vacillation in the spatial wavenumber of a packet. The author’s hypothesis (and this seems to be presented as the major idea in the monograph) is that the mechanism for this vacillatory behav- iour, namely suitable variation of the wavenumber following the packet, as a result of variations in the basic state, may be an explanation for more general vacillatory behaviour observed in experiment (e.g. the classic baroclinic annulus experiments) or in atmospheric observations. In Chapter 5 the author moves on to consider in detail the bifurcation structure of the wave packet behaviour in wavenumber space (as the variations in the basic state change). The basic results of bifurcation theory are reviewed as preparation. Some of these results are useful in the subsequent development, but sometimes the treatment seems formal and disconnected. For

928 Book reviews

example, Peixoto’s theorem is stated but curiously, when it is used in the subsequent chapters, it is not quoted. Chapter 6 considers the possibility of finding a sequence of bifur- cations in the wave packet system, though in fact only sec- ondary bifurcations are found. This material seemed no more than a thorough re-examination of the bifurcation structure already investigated in Chapter 5 and it was difficult to see why a separate chapter was needed.

In the final part of the book the author considers more realistic aspects of wave-packet behaviour. In Chapter 7 he extends the theory to three dimensions, first noting the proof of the Ertel potential vorticity theorem and working through the derivation of the quasi-geostrophic equations. The inves- tigation of wave-packet behaviour in three dimensions is not exhaustive by any means and in concluding the author merely notes that the extra degree of freedom might lead to chaotic behaviour in wavenumber space, but does not appear to have explored this idea seriously. Chapter 8 is concerned with the use of wave-packet ideas to interpret large-scale atmospheric Rossby wave patterns, particularly as manifested in tele- connection patterns. This is essentially a summary of work that was initiated in the early 1980s by Hoskins and Karoly on the theoretical side and Wallace and his collaborators on the observational side. However, the author neglects to mention any of the subsequent studies that have attempted to associate teleconnection patterns with large-scale insta- bilities of a longitudinally varying background flow rather than directly with forced Rossby wave trains.

As far as the introductory material goes this book does not really compare in thoroughness with standard texts such as Lighthill or Whitham. Most readers will therefore be primarily interested in the new results on wave packet behav- iour. The author argues that the vacillatory behaviour, in particular, is of wider significance and supports this in Chap- ter 4 by showing qualitative similarity between the vacillation in phase tilt of waves in a baroclinic annulus experiment and that predicted by the wave packet theory. However, the author goes no further in his comparison and in both the two subsequent chapters merely restates that there is qualitative similarity without offering any new supporting evidence. One specific problem is surely that whilst the wave packet theory predicts vacillation of the wavenumber following the packet, it also assumes that the packet is localised in space, whilst the vacillation observed in the laboratory, for example, appears to occur coherently through the whole flow domain. The author does not seem to explore whether a superposition of packets, localised at different points in the flow, can give rise to a coherent global vacillation in structure.

In conclusion, whilst the book has been carefully written, this reviewer is not convinced by its central argument. The author might have been better to take his own advice, given at the end of Chapter 4--‘Obviously further work is needed to test the prediction by using real data and making obser- vations’-and pursued the ideas further before publication.

Peter Haynes Department of Applied Mathematics and Theoretical

Physics, University of Cambridge

GaskineticTheory, Gombosi, Tamas, I., 1994,197 pp., Cam- bridge University Press, f40(hb), E17,95@b), ISBN O-521 43966-3 (pb).

It is a pleasure to read a book that leaves little doubt of the author’s thorough understanding of the subject on which he writes. Evidently, he is experienced in presenting the topic to

students. The reader is introduced very gently to gaskinetic- theory with a brief review of the development of the field, the physical assumptions on which kinetic theory builds and some elementary topological considerations. The concept of a distribution function is introduced early, setting the groundwork for the remainder of the text. A detailed deri- vation of the Maxwell-Boltzmann distribution is presented, and the properties which characterize the distribution are explored. Definitions of macroscopic properties of a gas are introduced as these appear in equations. Derivations are presented in the simplest form amenable to analytic solution and limited to classical microscopic interactions, although the limitations of classical theory and the importance of quanta1 treatments are pointed out.

In Chapter 3 binary collisions between molecules are anal- yzed. Starting with particle kinematics and intermolecular force fields the differential cross section is developed. The statistical properties of a gas, the collision frequency and the mean free path are derived in terms of the microphysical properties of collisions. The role of chemical reactions in binary collisions is briefly discussed.

Transport theory is introduced in Chapter 4 by allowing slow spatial variation of the macroscopic gas properties, and introducing concepts such as molecular diffusion and heat conduction. The goal of this chapter is to arrive at approxi- mate values for the transport coefficients. The assumptions in this treatment are clearly set forth and the limits of appli- cability are noted. The theory is applied to some elementary examples of gas flow in a tube, introducing terms that are extensively used in this field, the Mach number, the Reynolds number, and the Knudsen number. The author leads the reader through the steps of a derivation, and I recall reading only once the much abused phrase “it may readily be shown that ” (which frequently means after several pages of mathematical manipulations).

The Boltzmann transport equation is derived in Chapter 5. The flow terms (left side of the equation) are treated first, followed by the collision term on the right side. The H theorem is stated and proven, leading to an expression for the distribution function. It is shown that the Maxwelll Boltzmann distribution is the only equilibrium solution to the Boltzmann equation. Difficulties associated with eva- luating the collision integral analytically are noted, and approximate methods are derived, namely the relaxation lime approximation and the small velocity change approximation (Fokker-Planck). The chapter concludes by noting that in the real world non-equilibrium solutions frequently are required and that arriving at these solutions entails con- siderable mathematical complexity. The Chapman-Enskog method recognizes that transport phenomena occur only in gases that deviate from equilibrium, and that in a collision dominated gas departure from the equilibrium state is small and can be treated as a perturbation. The accuracy of the method depends on the order of expansion adopted but the results also depend on how well the differential cross section for collisions is known.

The first five chapters lead up to the meaty chapter 6 that deals with generalized transport equations. The method of Grad is introduced, i.e. taking velocity moments of the Boltz- mann equation. Closing the set of transport equations requires expressing higher order velocity moments of the distribution function in terms of components of lower moments. Yet, equations for the nth velocity moment also depend on components of the (n+l)th moment. The dilemma requires making assumptions about the distribution