Waves and mean flows by Oliver Bühler. Cambridge Monographs on Mechanics. Cambridge University Press, 2009. ISBN: 978-0-521-86636-1. 370 pp.

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The decomposition of a fluid flow into a mean part and a disturbance from that mean can be valuable both conceptually and as a framework for performing practical calculations. Conceptually, it provides a basis for our understanding in a wide range of situations. Examples of practical calculations include the representation of unresolved turbulence or wave momentum transport in numerical models. Prof. Bühler's book provides a comprehensive and up-to-date account of the theory of wave, mean-flow interactions. The term ‘mean flow’ is interpreted very broadly; it includes not only the more obvious zonal or time mean but also, in the final part of the book, large-scale flows that are sufficiently slowly varying compared to the wave field. The book is aimed primarily at application to atmospheric and oceanic flows, where the ubiquity of wave motions and the vastly multiscale nature of the flow make it especially relevant, but it is likely to be of interest for other fluid applications too.

The book is suitable for graduate-level students and beyond, and is likely to be of interest to both students and established researchers in geophysical fluid dynamics and other branches of fluid dynamics. It is assumed that the reader is comfortable with graduate-level mathematics, particularly vector calculus.

The book is structured in three parts, which systematically build up in complexity and depth. After a brief introduction to fluid dynamics, Part I covers the theory of propagation of linear waves on uniform and non-uniform backgrounds, including the techniques of WKB theory and ray tracing and the idea of caustics. The important ideas of wave action and wave activity conservation laws are introduced.

Part II describes the classic theory of wave–mean interaction, in which the mean flow is a zonal mean in a zonally symmetric geometry. The effects of shear and critical layers on the propagation of small amplitude waves are discussed. The pseudomomentum is introduced, and its crucial role in the feedback of waves on the mean flow explained. Generalized Lagrangian Mean theory is explained and shown to be a natural framework for the study of wave, mean-flow interaction, even for finite amplitude waves.

Part III of the book moves beyond the classic wave–mean interaction theory to discuss the interaction of waves with quite general slowly varying flows, without the restriction to mean flows and geometries possessing a simple symmetry. Much of this theory is recently developed, with significant contributions by Prof. Bühler himself. Here, a key quantity is the impulse of the mean flow, and under the right conditions the impulse (a mean-flow quantity) plus the pseudomomentum (a wave quantity) is conserved. The response of the mean flow to a wave field can be subtle and (at first glance) counter-intuitive. It can also be non-local: the effect of the waves on the mean flow can be felt at a location far from the wave packet, leading to the idea of remote recoil.

The book covers a great many topics beyond those mentioned above. The writing is always lucid and gets straight to the key ideas. The ideas are developed through well chosen examples, and at every step the mathematics is accompanied by clear physical interpretation. Several applications of the theory are mentioned, including understanding the stratospheric Quasi-Biennial Oscillation and the closure of the mesospheric jets, the robustness and self-sharpening of jets in shear flow, and the parametrization of gravity wave drag in weather prediction and climate models. The relevance of the theory to the development of numerical models is also mentioned in several places. I was left with the impression that the theory of waves and mean flows is not something that saw its heyday in the 1970's and 80's and is now moribund, but is alive and developing, as relevant now as it ever was.

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