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The first thing to be said about this book is that it is concerned with the analytical theory of turbulence at the deepest conceptual level and with mathematical analysis of a daunting complexity. Professor Leslie accepts that some may now question the need for such an arduous attack on the problem, bearing in mind the prospects of direct numerical analyses of ever increasing elaborateness, but hecontends that it is still necessary as he puts it, 'to tortureourselves like this'.In the context of practical meteorological problems the aim amounts, for example, to deriving the wind spiral in the boundary layer without recourse to the empirical device of eddy viscosity. In more general terms Professor Leslie's concern is with certain ideas and procedures, which have evolved since about the mid-I950s, for the more rigorous closure of the Navier-Stokes equations for turbulent flow. The closure difficulties arise from the fact that equations for the mean wind components involve second moments of these components (Reynolds stresses), and the equations for the second moments involve third moments, and so on, every additional equation bringing additional unknowns. The developments which are surveyed have sprung from the techniques of quantum field theory, in applications notably by R. H. Kraichnan, and to a large extent the book is an expounding and interpreting of the important advances made by Kraichnan and also by S. F. Edwards.The first three chapters provide a background of the established mathematics of turbulence and of the problem of closure. Chapter 4 then introduces the 'direct action' approximation which is the central theme of Kraichnan's approach, and the ideas are thereafter developed, partly in relation to alternative ideas for closure, in Chapters 5-12. 'Interaction' occurs between the modes of the wave numbers in the Fourier-transformed versions of the Navier-Stokes equationsand the adjective refers to the changes of energy in a certain wavenumber which arise directly from combinations of appropriately related wave numbers.Up to and including Chapter 12 the discussion is in terms of the idealised condition of homogeneous isotropic turbulence. In Chapter 13 the properties of channel flow as an example of the real flows studied by engineers, are discussed and in Chapter 14 the current methods of calculation, starting from the familiar eddy viscosity and mixing length models, are surveyed briefly. These lead in to the final Chapter (15) which deals with attempts made largely by Professor Leslie himself to bring the idea of 'direct interaction' to bear on real flows. It is interesting that the initial phase of Kraichnan's developments, which were couched in an Eulerian (fixed point) reference system, led in the late 1950s to properties of the turbulence spectrum which were inconsistent with Kolmogorov's theory and the famous minus-five-thirds power law. Kraichnan realised that the basic defect lay in the failure of the Eulerian specification to represent properly the sweeping round of small eddies large eddies...

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A diffusion approximation to inertial energy transfer in two-dimensional isotropic turbulence is derived in such a way that energy and squared vorticity are conserved. The −53 and −3 power inertial ranges are predicted.