11R1. Engineering Applications of Noncommutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups. - GS Chirikjian (Dept of Mech Eng, Johns Hopkins Univ, Baltimore MD) and AB Kyatkin. CRC Press LLC, Boca Raton FL. 2001. 674 pp. ISBN 0-8493-0748-1. $89.95.

Reviewed by AC Buckingham (Center for Adv Fluid Dyn Appl, LLNL, Mail Code L-23, PO Box 808, Livermore CA 94551).

This is a monograph, providing a review and summary of the author’s original causality concepts and research on the origin and evolutionary development of semi-organized, large-scale, energetic flow structures. These emerge from very much smaller scale unstable, transitional, and ultimately, statistically-disordered turbulent flows. Apparently intended for the inquisitive specialist, it is written with emphasis on sophisticated exposition rather than instruction. However, physics, mathematics, and engineering science graduate students with an interest in the on-going developments, revelations, and consequent inevitably deeper questions about the turbulent motion of fluids should find it a useful source of stimulation for discussion and (perhaps) greater understanding. It should also be of interest to senior specialists: those investigating ecological fluid dynamics with attention to the spread of pollutants; those investigating atmospheric fluid dynamics with attention to catastrophic storm formation and climate change; those investigating and developing drag and heating reduction techniques for aerodynamic applications; those investigating film and blanket cooling in nuclear and thermal power generation systems; and those investigating systems dominated by magnetically coupled plasma flows and magneto hydrodynamics (MHD) generally.

The book represents an exposition of three or more decades of individual and later collaborative investigation by an innovative team from Israel and Russia with a shared special interest and emphasis in plasma flow processes, instability development, and the influence and development of large scale, quasi-coherent flow structures in transitional and turbulent flow. Specifically, the material presented is developed from research pursued in the Joint Israeli-Russian Lab for Energy Research, Ben Gurion University at Beer-Sheva, Israel in collaboration with that of the Cosmogeophysics Department of the Institute for Space Research, Moscow, Russia.

The presentation is expository. The language and text organization provide clear descriptions and a logical development of the connected themes. Concepts are described physically and in analogy with commonly experienced observations of flow behavior and the apparent, often directly measurable, physical consequences. Mathematical relations are introduced that are effectively selected for use in illustrating physical inter-relationships rather than for formal derivation and proof construction. For example relationships are developed and used to describe analysis of the conditional changes which may be anticipated to strongly affect the outcome of an evolutionary process; the altered behavior anticipated at early and late system time limits; the altered behavior anticipated for very large and very small limit values of system parameters; and the isolation, identification, and causal consequences of alterations in flow properties.

At least two of the central themes of the book deserve special comment. They are complimentary. The first is the helicoidal large scale 3D flow structure which is energized and developed as a consequence of the second, a reversal in the direction of transfer of energy over the inertial range of the energy spectrum. The spiraling helical trajectories form the framework of the helicoidal 3D structures. The largest scale size quasi-coherent structures contain most of the kinetic energy of the flow in direct analogy with the largest plane vorticity structures in 2D turbulence. These 3D helicoidal structures appear as almost stationary, quasi-periodic background features. In helicoidal dynamics, the conserved quantity is helicity. Helicity is expressed as the scalar product of the velocity and vorticity vectors and is a conservative quantity in inviscid, non-dissipative flows. Helicity is also considered analogous to the conserved quantity, exstrophy (mean squared vorticity) in 2D turbulence. While the helicoidal structures unquestionably exist, the dynamics of their evolution, the dominant role they may play as energy sinks and dissipation sources remain the focus of considerable debate in the turbulence community. Helicoidal topological features and concepts of their evolution, intermediate states, and dissipation have been extensively studied in the well-known work of E Levich and colleagues, and HK Moffatt and colleagues in the 1980s and early 1990s.

We now focus special attention on a second concept, the reversal of the flow of energy from the smallest dissipative scales to the largest production range scales. The well established conventional inertial range cascade of energy from large to small scale, developed from the statistical analysis of Kolmogorov and Obukhov in the 1940s has been confirmed in numerous experiments and in direct numerical simulations. Its existence is not at issue. What is proposed in this book (and in some other publications) is this may not be the only significant process and perhaps not even the dominant process governing every form of turbulent energy transfer and transition of states. The suggestion is advanced in this book that this reversal of the inertial range energy transfer process, observed in many flow situations, may dominate the production, development, and duration of all scales of motion from the near-coherent to the statistically random.

The reversed energy cascade whether dominant or peripheral in turbulence dynamics has also been established early in the 1960s in the United States in the context of what was for a time a paradoxical and unconfirmed state of “2D turbulence.” Despite experimental evidence (from what is the largest size scale existing experimental “facility,” the planetary atmosphere) publications involving the concept were rejected. The log-jam of publications may have terminated with a “break-through” paper on the structure of 2D turbulence by the distinguished fluid dynamicist, GK Batchelor in the 1960s. Experimental evidence was forthcoming from atmospheric measurements showing the existence of a −3 power law dependence for the energy spectrum which corresponds uniquely to a 2D turbulence state just as the −5/3 power law dependence corresponds to the 3D turbulence state established by the work of Kolmogorov-Obukhov. Of course, this essentially 2D character or “flat” planar character of atmospheric turbulence is readily apparent from the observation that structural flow changes in the vertical atmospheric direction occur on a scale orders of magnitude smaller (hence much more rapidly) than the analogous changes in the horizontal direction. The noted fluid dynamicist and a recognized leader and originator of many of the fundamental contemporary concepts in theoretical turbulence, RH Kraichnan, in a well-known 1967 paper suggested a very compelling analogy in explanation of this inverse cascade process also known as “stochastic back scattering.” He pointed out that in a “flat” 2D world, the disk-like vortices interacted in an analogous way to Bose-Einstein condensates and that under the constraints of the 2D conservation of enstrophy would necessarily, organize themselves into larger and larger structures which would ultimately coalesce into a single vortex structure encompassing the entire system (in the absence of dissipation).

Naturally occurring as well as laboratory experimental observations are a positive and most welcome addition to the book’s content. Experimental observations discussed cover a usefully broad range of flow situations; among them: the onset and evolution of cyclones and tropical typhoons in atmospheric fluid dynamics; earth mantle thermal-fluid convection current circulation in planetary seismological and terrestrial magnetic field studies; and results of artificially stimulated large scale structure formation in applied aerodynamic schemes for turbulent drag and heat transfer reduction.

With the exception of their appearance in sections presenting and interpreting experimental results, the number of figures is modest. This, however, is not a drawback in the presentation because the carefully-prepared text deliberately stimulates the reader’s curiosity, provides for the reader an appreciation of the author’s perspective, and brings into sharp focus identification of the issues together with the authors’ interpretations and conclusions.

There is a substantial list of over 310 references. This includes reference to much of important original work on nonlinear instability development in plasma and neutral gas flow carried out in the USSR in the 1960s and 1970s as well as the authors’ own considerable contributions spanning essentially the last three decades. There is a more modest, but selectively representative reference to the associated published research accomplished in Britain, France, and the US during the same era. There is also a comprehensive, if compact, cross index of nearly 500 topics and subtopics which the reader will find useful as a guide for pursuing background source information in the text and in related references. A novel feature is the compact disk which is storage for the reference data list and can be found in a pocket attached inside the book’s back cover. This can be downloaded onto the reader’s computer. It is designed to be used as an interactive author/subject data information-and-storage list that can be modified and added to at will by the reader.

Sequentially, the book is arranged into seven chapters. Experimental observations frequently enhance the discussion. The first chapter contains introductory material on stability analysis for continuum flow and defense of the utility of linear stability analysis in mapping out stability limits for non-linear systems. The book then quickly takes the reader to a description of both primary, dissipative instabilities, induced secondary instabilities, and the eventual dynamical path to formation of large scale helicoidal structures. Dynamic systems concepts are introduced next to assist description of the way in which almost stationary flow structural states may emerge from systems with limited degrees of freedom (ie, unstable or sub-transitional flow states). Experimental examples follow, including natural thermal-fluid instability induced geophysical convection currents as well as the formation of the early stages of catastrophic weather systems such as typhoons and tropical cyclones. The reader is next given an introduction to statistical fluid dynamic concepts statistical correlations and higher moments in describing cross coupling which is proposed as a mechanism for self-rearrangement and organization of larger and larger flow structures. Next experimental evidence is presented illustrating the influences of magnetically coupled generation and growth of the large-scale helicoidal structures in measured changes to drag and heat transfer in duct flow, and evidence on the forced changes to persistence of the structures in grid generated turbulence. Here the authors provide some compelling arguments about the importance of laboratory scale experiments such as these to assist in the interpretation and description of much larger scale naturally occurring geophysical and atmospheric flows. They finish with a summary with additional albeit limited experimental evidence placing emphasis on the influences of the large scale forced flow structural modifications on reducing unwanted heat and mass transfer in contained flows and an exposition of the boundary condition influences in these systems.

Engineering Applications of Noncommutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups should be a source of enlightenment to the reader on many hitherto unexplored issues and concepts connecting the production and dissipation ranges of turbulence and the unstable developmental phases leading to these states. The authors have succeeded in producing a very readable and informative book that should act to stimulate additional discussions, investigations, and perhaps a little more progress and understanding of the turbulence problem. It is recommended for research laboratory and academic library acquisition as a novel source of important developments. For the interested reader, however, probably nothing but a personal acquisition will suffice because it invites re-reading at intervals to appreciate the important, but often controversial concepts that are introduced.