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Energetic Consistency and Momentum Conservation in the Gyrokinetic Description of Tokamak Plasmas

Gyrokinetic field theory is addressed in the context of a general Hamiltonian. The background magnetic geometry is static and axisymmetric, and all dependence of the Lagrangian upon dynamical variables is in the Hamiltonian or in free field terms. Equations for the fields are given by functional derivatives. The symmetry through the Hamiltonian with time and toroidal angle invariance of the geometry lead to energy and toroidal momentum conservation. In various levels of ordering against fluctuation amplitude, energetic consistency is exact. The role of this in underpinning of conservation laws is emphasised. Local transport equations for the vorticity, toroidal momentum, and energy are derived. In particular, the momentum equation is shown for any form of Hamiltonian to be well behaved and to relax to its magnetohydrodynamic (MHD) form when long wavelength approximations are taken in the Hamiltonian. Several currently used forms, those which form the basis of most global simulations, are shown to be well defined within the gyrokinetic field theory and energetic consistency.