Generalized Hamiltonian drift-fluid and gyrofluid reductions

We provide a procedure for deriving Hamiltonian reduced fluid models for plasmas, starting from a Hamiltonian gyrokinetic system in the delta f approximation. The procedure generalizes, to a considerable extent, previous results. In particular, the evolution of moments with respect to the magnetic moment coordinate is also taken into account, together with background density and magnetic inhomogeneities. In the limit of vanishing Finite Larmor Radius (FLR) effects, an infinite family of reduced electron drift-fluid equations is derived, evolving all the electron moments g(ije), with i = 0,..., N and j = 0,..., M, where N and M are arbitrary non-negative integers counting the maximum order of the moments taken with respect to the parallel velocity and to the magnetic moment coordinates, respectively. An analogous result is found in the gyrofluid case and applied to the ion species. The gyrofluid result holds for M <= 1, finite FLR effects and a low ratio ss e between electron internal pressure and magnetic guide field pressure. In both the drift and gyrofluid case, the key for the identification of the Hamiltonian structure resides in changes of variables based on orthogonal matrices that diagonalize the Jacobi matrices associated with Hermite and Laguerre polynomials. In terms of the transformed variables, the drift and gyrofluid equations are cast in a simple form, which reduces to advection equations for Lagrangian invariants in the two-dimensional case with homogeneous background. Because the procedure requires to evolve, for a particle species s, all and only the moments g(ijs), with i = 0,..., N and j = 0,..., M, not every choice for the set of moments is admissible, for fixed N and M. This might also explain the scarcity of Hamiltonian reduced fluid models obtained so far which account for anisotropic temperature fluctuations.