Poisson brackets and truncations in nonlinear reduced fluid models for plasmas
Résumé
The Hamiltonian structure for an infinite class of nonlinear reduced fluid models, derived from a Hamiltonian drift-kinetic system, is explicitly provided in terms of the N + 1 fluid moments evolving in each model of the class, with N an arbitrary positive integer. This improves previous results, in which the existence of the Hamiltonian structure was shown, but the complete explicit expression for the Poisson bracket of each model of the class was not provided. We also show that, whereas the Hamiltonian functional of the fluid models can be derived from that of the drift-kinetic system, by projecting the perturbation of the distribution function onto its truncated series in terms of Hermite polynomials, this is not the case for the Poisson bracket. Indeed, the antisymmetric bilinear form obtained by means of the aforementioned projection, although, interestingly, "very similar" to the Poisson bracket of the fluid models, turns out to differ from it. The difference is found to reside in the coefficients W(N)lmn of the bilinear form, when the indices are such that l + m + n is even and l ≥ N + 1 , m ≥ N + 1 , n ≥ N + 1 . We show with a counterexample, related to the case N = 2 , that such bilinear form, in general, does not satisfy the Jacobi identity. We provide a physical interpretation of the set of variables G0 ,G1 , … ,GN , in terms of which the Poisson bracket of the fluid models exhibits a direct-sum structure, and point out an analogy between the present fluid reduction problem and the problem of the truncated quantum harmonic oscillator.
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