Magnetohydrodynamic Turbulence

While hydrodynamic turbulence has been a major subject of physical research for more than a century, turbulence in plasmas is unusual in daily life or even laboratory conditions, and its relevance only became apparent with the development of astrophysics. For macroscopic processes, the magnetohydrodynamic description is usually adequate, and its application to realistic phenomena started with the work of Batchelor in 1950 on the dynamo problem. Solar flares, the solar wind, the geodynamo and stellar accretion disks, to name only a few, were added later to the list of phenomena amenable to a MHD description: naturally the literature on such a vast subject is immense, but so far there was not a reference book providing an up-to-date account of the main concepts and results. The monograph by Dieter Biskamp is well suited to fill this void. The MHD equations are similar in many ways to the Navier-Stokes ones, and the parallelism extends to several key concepts in the understanding of turbulence, such as self-organization, cascades and closure methods. Perhaps the strongest part of the book from a didactic viewpoint is the way this parallelism is exploited to highlight the similarities and differences with hydrodynamic turbulence, and the form that key phenomena exclusive of MHD, such as the Alfvén effect or flow anisotropy, modify the classical results. After a clear and concise introduction to the basic MHD equations, ideal invariants and linear waves, the book describes some classical instabilities leading to turbulence, such as the Kelvin--Helmholtz instability. This is followed by the statistical theory of incompressible turbulence. As the author asserts, the dynamical systems approach pioneered in the nineteen eighties has failed to produce new physical insights, not because of any intrinsic flaw, but because of the large number of degrees of freedom (in mathematical parlance, essentially the dimension of the attractor) present in turbulent fluids and plasmas. We must therefore handle three essential methods: phenomenological scaling arguments in the spirit of Kolmogorov's K41 theory; closure theories, obtained by truncating at some point the hierarchy of moment equations; and to verify the plausibility of these approaches, numerical simulations of the original dynamic equations. Biskamp is a master of numerics and the book is well-illustrated with graphics pointing out the strengths and shortcomings of these theories. The key role of two specific MHD invariants (the magnetic helicity and the cross-helicity) in the cascades direction is very clearly explained, as well as the ranges of applicability of the Kolmogorov and the Iroshnikov-Kraichnan statistics, about which there was some polemic until recently. Compressible turbulence and turbulent convection is tackled next, and the monograph ends with studies of three specific astrophysical topics: the solar wind, accretion disks and interstellar turbulence. Here the text becomes of necessity more descriptive and empirical, as the complexity of the phenomena grows in inverse proportion to our detailed knowledge of them. Nevertheless some of the more amenable processes, such as the instabilities of certain geometries, are reasonably detailed and on the whole one gets the feeling of understanding the basics of the problems. Few criticisms can be levelled at this monograph, and most of them are answered by the need to keep its length within bounds. Thus, turbulent dynamos and turbulent reconnection are almost entirely omitted, and other subjects, such as mean-field electrodynamics and decay laws, are more controversial than the author admits. Also, while Biskamp is probably right in being skeptical about deriving much knowledge on intermittency from the dynamic equations alone, some useful estimates on scaling exponents have been rigorously proved, beginning with the work of Constantin and Fefferman (1994). This certainly does not detract from the excellent global impression obtained from this book, which undoubtedly belongs on the shelves of every student of MHD turbulence. M Núñez