Response to “Comment on ‘Guiding center plasma models in three dimensions’ ” [Phys. Plasmas 16, 084701 (2009)

Abstract

In response to the Comment [J. A. Krommes, Phys. Plasmas 16, 084701 (2009)], this note extends the original paper [L. E. Sugiyama, Phys. Plasmas 15, 092112 (2008)] to consider the breakdown of good magnetic flux surfaces. The validity of the noncanonical Lagrangian (NCL) guiding center model of Littlejohn and Brizard depends directly on the existence of sufficiently smooth, structured solutions for the three-dimensional magnetic field in a plasma. In general, such fields are chaotic. Local existence of the parameter R=(∇ê1)⋅ê2, which measures the rotation of the magnetic coordinate system for the gyroangle, determines the existence of the NCL expansion at second order. Globally, sufficient smoothness that R is definable almost everywhere requires zero magnetic field line torsion or two-dimensional (planar) symmetry, although limited regions of validity may exist otherwise. The original anholonomy analysis of Littlejohn for the gyroangle assumes field smoothness and concludes that R almost always exists.