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Hamiltonian fluid reductions of drift-kinetic equations and the correspondence with water-bag distribution functions

Hamiltonian models for the first three moments of the drift-kinetic distribution function, namely the density, the fluid velocity and the parallel pressure, are derived from the Hamiltonian structure of the drift-kinetic equations. The link with the water-bag closure is established, showing that, unlike the one-dimensional Vlasov equations, these solutions are the only Hamiltonian fluid reductions for the drift-kinetic equation. These models are discussed through their equations of motion and their Casimir invariants.