Nonlinear theoretical tools for fusion-related microturbulence: Historical evolution, and recent applications to stochastic magnetic fields, zonal-flow dynamics, and intermittency

Abstract

Abstract. Fusion physics poses an extremely challenging, practically complex problem that does not yield readily to simple paradigms. Nevertheless, various of the theoretical tools and conceptual advances emphasized at the KaufmanFest 2007 have motivated and/or found application to the development of fusion-related plasma turbulence theory. A brief historical commentary is given on some aspects of that specialty, with emphasis on the role (and limitations) of Hamiltonian/symplectic approaches, variational methods, oscillation-center theory, and nonlinear dynamics. It is shown how to extract a renormalized ponderomotive force from the statistical equations of plasma turbulence, and the possibility of a renormalized K-χ theorem is discussed. An unusual application of quasilinear theory to the problem of plasma equilibria in the presence of stochastic magnetic fields is described. The modern problem of zonal-flow dynamics illustrates a confluence of several techniques, including (i) the application of nonlinear-dynamics methods, especially center-manifold theory, to the problem of the transition to plasma turbulence in the face of self-generated zonal flows; and (ii) the use of Hamiltonian formalism to determine the appropriate (Casimir) invariant to be used in a novel wave-kinetic analysis of systems of interacting zonal flows and drift waves. Recent progress in the theory of intermittent chaotic statistics and the generation of coherent structures from turbulence is mentioned, and an appeal is made for some new tools to cope with these interesting and difficult problems in nonlinear plasma physics. Finally, the important influence of the intellectually stimulating research environment fostered by Prof. Allan Kaufman on the author's thinking and teaching methodology is described.

A tutorial introduction to the statistical theory of turbulent plasmas, a half-century after Kadomtsev’sPlasma Turbulenceand the resonance-broadening theory of Dupree and Weinstock

“…One can therefore use a functional chain rule: 27) where (3.24) was used in the last step. This formally closes the equation for R in the form of a Dyson equation (Dyson 1949;Krommes 2002Krommes , 2009):…”

“…The basic point is that the naive quasi-linear result for Σ scales with β 2 , but a self-consistent calculation of the nonlinear damping rate for large K and R leads to a scaling with β 1 ; here the power 1 is called an anomalous exponent. Introductory pedagogical discussion of renormalization and anomalous scaling can also be found in the article by Krommes (2009), which provides some useful background for the present tutorial.…”

“…= ∂δf · E E E that defines the nonlinear corrections to the background distribution function in the renormalized dielectric, the physical interpretation of those terms is still largely absent for kinetic problems (mostly for lack of trying, although see DuBois & Pesme (1985)). A hint comes from the work of Krommes (2009), where it was shown how the notions of oscillation-centre variables and ponderomotive force are buried in the renormalizations. An outstanding question is whether such analysis can be extended to δf .…”

Bob Dewar and turbulence theory: lessons in creativity and courage

“…It is interesting that a (partial) answer to this question was only given in 2007, some thirty years after Dewar's work on renormalized oscillation-center theory. In the proceedings of the symposium honoring Allan Kaufman's 80th birthday, the articles for which may be found online (Brizard and Tracy 2009) 9 , Krommes (2009) carefully worked out the mass operator for renormalized, inhomogeneous quasilinear theory, from which he was able to extract the usual ponderomotive force. However, whereas that force follows so neatly from oscillation-center theory, it is somewhat buried in tedious algebra in the MSR approach.…”

The physics of the second-order gyrokinetic magnetohydrodynamic Hamiltonian: μ conservation, Galilean invariance, and ponderomotive potential

“…17, and the attempt at a renormalized K-v theorem in Ref. 18.) The applicability of this wave-based formula to the physics of a gyrating particle requires further discussion but can be heuristically justified by drawing an analogy between (i) time averaging over the fast quiver of a particle in a wave field, and (ii) averaging over the rapidly changing gyration phase of a particle in a magnetic field; cf.…”