Transport and Structural Formation in Plasmas

Abstract

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 involved, including the crucial ones of random phases and the applicability of radial Fourier expansions in a manifestly inhomogeneous system. After all the formidable algebraic formalism, in practice, the actual formulas boil down to a simple `mean field' prescription which is only marginally more realistic than the simple mixing length model of Chapter 4. It is clear that renormalization theories often miss important features such as turbulent advective transport and the damping effects of linearly stable modes of the spectrum (such as shear Alfvén or flow `continua') on unstable modes. One would expect these caveats to have been more thoroughly and critically discussed in a monograph such as this.The next few chapters apply these ideas to particular models of turbulence favoured by the authors. The results, though interesting, are not entirely convincing. There are no comparisons with experiment. The simulations are done with an electrostatic model in which the equilibrium pressure gradient is fixed, rather than fixed energy and particle sources and a self-consistent pressure evolution. In reality, flux surface averaged turbulent fluxes have rapidly varying radial/temporal behaviour, which feeds back to the equilibrium (i.e. G0 varies with x, t on the mesoscale) and hence the turbulence itself. Simulations with fixed gradients can be seriously misleading indicators of actual turbulence structures on the mesoscale.The rest of the book (Chapters 16-24) is devoted to electric field effects and transport barriers. While there is much that is interesting in what the authors have to say here, the treatment is cavalier in places and insufficiently rigorous from the physical point of view. The authors explain the effects due to radial electric field shear on instabilities in terms of a reduction in the perpendicular wavelength (Section 18.2). This amounts to a shear flow induced direct cascade to higher perpendicular wavenumbers. This is a purely linear process analogous to `phase mixing' and may also be understood as a form of `continuum damping' (not discussed in this book) due to the E × B flow. More subtle non-linear mechanisms involving the electric field also exist, but these are only hinted at. The real difficulties with the book arise in what determines the fields themselves. The authors make the statement that radial electric fields are associated with radial currents (Chapter 19) and seek to justify this with a model (Eq. (19.2)). Unfortunately, the model is flawed, since the friction forces chosen are far too special to be representative of the full equations. In a quasi-neutral plasma subject to low frequency (i.e. ω<<ωpe) turbulence, the displacement current is always legitimately negligible and Ampère's law shows that on any internal closed flux surface, the total integrated ion and electron fluxes must be exactly equal (i.e. there cannot be any net accumulation of charge inside such surfaces to (λD/R)). Both toroidal and poloidal flows within such flux surfaces are determined by balancing the corresponding stresses against applied sources of momentum (if any). Thus, for example, in classical Braginskii theory in a periodic cylinder model, in the absence of external momentum inputs, the ion in-surface flows nearly vanish and do not contribute to the radial electric field which is determined essentially by the ion pressure gradient (missing in Eq. (19.2)). Furthermore, the radial particle flux is related to the poloidal current density via a generalized Ohm's law/electron momentum balance. These results are exact in this model and clearly form a counterexample to the claims of the authors based on their inadequate momentum balance equation mentioned above. The authors use `current' (e.g. Eq. (19.6-1)) to mean particle flux of a particular species and appear to ignore the fact that friction forces also depend on relative flows between different charged species. Thus an oversimplified model gives rise to several conclusions which are misleading and may be invalid under relevant experimental conditions. The final chapter contains interesting material on the internal transport barriers. However, the descriptions of the transport equations are inadequate and opaque. Not discussed are issues relating to why low rational values of q seem to be important to the formation of such barriers, although the authors do note the importance of local bootstrap currents generated by the large pressuregradients.The text reads ambiguously or awkwardly at many points, and would have benefitted from careful editing with special reference to acceptable English style. A general problem is the tendency to sweep away crucial points under the carpet with references to the literature. This may be a reasonable strategy for computational niceties or tedious algebra, but surely not for matters of principle in what is supposed to be a self-contained monograph.This book has some attractive features which may make it a useful reference for its intended audience and help sweep away certain persistent a priori misconceptions about the role of linear analysis in turbulence theory. As an authoritative monograph however, it fails to carry conviction, at least to this reader.