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

Abstract

Some physical interpretations are given of the well-known second-order gyrokinetic Hamiltonian in the magnetohydrodynamic limit. Its relations to the conservation of the true (Galilean-invariant) magnetic moment and fluid nonlinearities are described. Subtleties about its derivation as a coldion limit are explained; it is important to take that limit in the frame moving with the E  B velocity. The discussion also provides some geometric understanding of certain well-known Lie generating functions, and it makes contact with general discussions of ponderomotive potentials and the thermodynamics of dielectric media. V C 2013 AIP Publishing LLC.It is well known that gyrokinetics (see the review by Krommes 1 and references therein, as well as the pedagogical material in Ref. 2) can be described as a Lagrangian field theory. 3,4 In full generality, including gradients of the magnetic field B and finite-Larmor-radius (FLR) effects, the gyrocenter Hamiltonian H is quite complicated, even at second order 5 in the gyrokinetic expansion parameter . However, in the absence of gradients of B and FLR effects, the second-order Hamiltonian simplifies dramatically to the well-known form 6-9where M m i is the ion mass and u E ¼ : cb  $/=B is the E  B velocity. (I consider only electrostatics, with / being the electrostatic potential such that E ¼ À$/; the constant magnetic field is written as B ¼ Bb.) Authors such as Scott and Smirnov 10 have used this "magnetohydrodynamic (MHD) Hamiltonian" to good advantage in order to illustrate various aspects of gyrokinetics such as conservation properties. Clearly H 2 is the negative of the kinetic energy associated with the E  B motion of the gyrocenters; the reason for the minus sign may not be immediately apparent. In this note I give several relatively elementary derivations of H 2 and discuss some subtleties that may not be generally appreciated. An excellent review of modern algebraic results is given in Ref. 9; here I focus on physical interpretations that have not appeared or been sufficiently emphasized in the literature.In particular, note that the MHD Hamiltonian (1) does not involve the ion temperature T i ; it remains nonzero in the cold-ion limit T i ! 0. Frequently the program for obtaining a gyrokinetic Hamiltonian is described as a systematic, order-by-order elimination of gyrophase f, which inevitably conjures up the idea of an average over a rapidly rotating gyroradius vector q, where q ¼

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