A Quarter-Century Later: Nonlinear Gyrokinetics Under Attack

The nonlinear gyrokinetic equation (GKE) was first derived about a quarter-century ago. Subsequent technical developments have refined the GKE into a major tool for the description of both fusion and astrophysical plasmas. However, the GKE has suffered serious attacks on its veracity, two of which will be discussed: the possibilities that (i)~the asymptotic expansion for the GK variables breaks down for torsional or stochastic magnetic fields\footnote{L. E. Sugiyama, Guiding center plasma models in three dimensions, Phys.\ Plasmas \textbf{15}, 092112 (2008); J. A. Krommes, Comments on ``Guiding center plasma models\dots'' [Phys.\ Plasmas \textbf{15}, 092112 (2008)], Phys.\ Plasmas (2009, submitted); L. E. Sugiyama, Response to Comments of J. A. Krommes, Phys.\ Plasmas (2009, submitted).}; (ii)~conventional gyrokinetics is insufficiently accurate to determine the long-wavelength, axisymmetric part of the radial electric field.\footnote{F. I. Parra and P. J. Catto, Limitations of gyrokinetics on transport time scales, Plasma Phys.\ Control.\ Fusion \textbf{50}, 065014 (2008).} The relevant physical pictures and detailed mathematics will be described for both sides of each issue. For (i), local and global coordinate systems must be distinguished; for (ii), the use of Lagrangian field theory is advocated.