Statistics – Computation
Scientific paper
Jun 1999
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1999nucfu..39..821t&link_type=abstract
Nuclear Fusion, Volume 39, Issue 6, pp. 821-823 (1999).
Statistics
Computation
1
Scientific paper
The book under review is one of a series of monographs on plasma physics published by the Institute of Physics under the editorship of Peter Stott and Hans Wilhelmsson. It is nicely produced and is aimed at research workers and advanced students of both laboratory (i.e. tokamak plasmas) and astrophysical plasma physics. The authors are prolific contributors to the subject of plasma turbulence and transport with a well-defined message: ``The authors' view is that the plasma structure, fluctuations and turbulent transport are continually regulating each other and, in addition, that the structural formation and structural transition of plasmas are typical of the physics of far from equilibrium systems. The book presents and explains why the plasma inhomogeneity is the ordering parameter governing transport and how self-sustained fluctuations can be driven through subcritical excitation even beyond linear instability''.
This point of view is expounded in 24 chapters, including topics such as transport phenomena in toroidal plasmas (Chapters 2-4), low frequency modes and instabilities of confined systems (Chapters 5-7), renormalization (Chapter 8), self-sustained turbulence due to the current diffusive mode and resistive effects (Chapters 9-11), subcritical turbulence and numerical simulations (Chapters 12-14), scale invariance arguments (Chapter 15), electric field effects (Chapters 17-21) and self-organized dynamics (Chapter 22). The material is essentially drawn from the authors' many and varied original contributions to the plasma turbulence and transport literature.
Whatever view one might have about the merits of this work, there is little doubt in this reviewer's mind that it is indeed thought-provoking and presents a worthy intellectual challenge to plasma theorists and experimentalists alike. The authors take a consistent stance and discuss the issues from their own standpoint. They observe that the plasmas one encounters in practice (for definiteness, the tokamak can be taken as an illustrative example) are clearly dissipative open systems, which are invariably driven far from thermodynamic equilibrium by means of a suitable set of external particle, momentum, energy and current sources. In this sense, such plasmas are analogous to the Earth's atmosphere and many other fluid dynamic systems one encounters in engineering and physics. It is well known that the transport processes in such systems are describable by strictly collisional, kinetically derived models such as neoclassical theory or laminar fluid flow equations only in exceptional circumstances. The generic case is one in which the system acquires `structure' in the sense that symmetry-breaking spatio-temporal turbulent micro/mesoscale fluctuations `spontaneously' occur, and in their turn influence the macroscale evolution of the system. Thus, given typical values of density, temperature, magnetic field and current, the tokamak plasma does not automatically reach a steady state consistent with the sources, symmetry and neoclassical equations. Rather, one finds a more or less turbulent state which often (but not always!) involves much worse thermal and particle insulation than expected on the grounds of Coulomb collisional processes alone. The authors seek to promulgate a particular model which does not require the existence (in principle) of any linear instability of the `equilibrium'. This is a well-known state of affairs in fluid dynamics (e.g. pipe flow) when turbulence can occur in spite of the fact that linear theory predicts the equilibrium to be stable.
While this is indeed a welcome clarification of the relatively limited role of linear theory in describing plasma turbulence in any detailed predictive sense, it is not clear why the authors elevate `subcritical turbulence' to a fundamental principle. While it may well be present, it is in general neither necessary nor sufficient to explain turbulent transport in plasmas.
In this reviewer's opinion, at the heart of the problem is a set of non-linear equations (fluid or kinetic) describing electromagnetic turbulence. These have linearly or non-linearly unstable steady solutions for experimental conditions of interest. The instabilities are invariably non-linearly saturated at sufficiently high fluctuation amplitude, resulting in approximately stationary (but not generally homogeneous or isotropic) turbulence. The turbulence, as a rule, tends to increase the radial transport of density, temperature, momentum and current (for given sources) and thereby lower their gradients. The authors call such gradients `order parameters' in analogy with condensed matter phenomenology (Ginzburg-Landau theory). Whatever one calls them, if the growth rates of turbulent plasma modes are proportional to such gradients (which they are in simple cases of linear instabilities), one has available a `self-organizing' feedback loop. If this is all that the authors wanted to say, it is indeed unexceptionable and they should be applauded for pointing out that a clear conceptual understanding of these ideas is independent of any kinetic complications and reservations about so-called `quasi-linear estimates' of confinement which permeate the subject.
Unfortunately, however, what is merely an illustrative model seems to be taken to empirically untenable and theoretically unjustifiable extremes. For example, we are told rather grandly that the book addresses ``a key to understanding the age old question of what occurred in the early stages of our universe and what is likely to occur in the final stages of our universe''. It is hard to discover where this feat is accomplished in this book! It is not clear that the models used by the authors, such as the `current diffusive mode', are really at all relevant to actual tokamaks. If they are, many more predictions (not postdictions) of the model should be made and verified in detail experimentally. At best, all one can say is that such models may not be inconsistent with experiment. The authors can be congratulated for illustrating the potentialities and promise of certain paradigmatic two fluid models of tokamak turbulence and transport. But this is very far from claiming at present that a future theory of tokamak transport will be based solely, or even essentially, on similar ideas. The demonstration of a possibility should not be interpreted as a claim of validity.
The early Chapters 1-7 give a brief survey of the basic notions of confinement and transport in toroidal systems. The material is well presented and should be accessible to most students. References are provided to the literature for deeper treatments. The treatment of equilibrium and neoclassical transport is rather superficial and the uninitiated reader may find many of the arguments obscure. The authors miss a useful opportunity to discuss in this context such issues as the limitations of the so-called `electrostatic' limit of the reduced equations in the neighbourhood of mode rational surfaces and the `automatic' ambipolarity of turbulent particle transport. The system is governed by the reduced equations which, after all, use ∇ ·j = 0 (also called the `shear Alfvén law' or vorticity equation) as one of the dynamical evolution equations. While it is natural in a monograph of this kind that the authors should seek to further their views, it is unfortunate that they appear to persistently avoid discussion of alternate hypotheses or approaches which may be counter to their position. This tendency for selective citation leads to rather one sided presentations of ideas in various places.
Chapter 8 takes up the concept of `renormalization'. It is doubtful if the discussion there will make much sense to a non-expert. The idea itself is quite old and traceable to Kramers' work in quantum field theory and to Prandtl and Kolmogorov in fluid mechanics. It simply embodies the fact that the non-linear terms in the equations of motion can sometimes be approximated in terms of `effective' diffusivities which depend on the solutions in some self-consistent manner. There are many ad hoc assumptions
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