Derivation of Hamiltonian magnetofluid models with gyroviscous-like contributions using a gyro-map

Ideal MHD and various reduced magnetofluid models exhibit a noncanonical Hamiltonian structure when expressed in Eulerian variables [1]. Extending the work of Ref.[2], we investigate the possibility of systematically including contributions due to finite ion gyro-radii in three dimensions while preserving a noncanonical Hamiltonian structure. Starting with the Morrison-Greene 3D ideal MHD noncanonical Poisson bracket[1] and a Hamiltonian including gyroviscous terms, we derive equations of motion using a three-dimensional generalization of the gyro-map introduced in Ref.[2]. The origin of the gyro-map is motivated and explained using an action principle formulation as in Ref.[3]. Through a systematic reduction procedure, we also recover the (noncanonical) bracket and the gyroviscous tensor, which are identical to the ones obtained via the Hamiltonian formalism. \\[4pt] [1] P.J.~Morrison and J.M.~Greene, Phys. Rev. A {\bf 45},790 (1980).\\[0pt] [2] P.J.~Morrison, I.L.~Caldas, and H.~Tasso, Z. Naturforsch. {\bf 39a}, 1023 (1984).\\[0pt] [3] P.J.~Morrison, M.~Lingam, and R.~Acevedo, arXiv:1405.2326 (to appear in Phys. Plasmas)