https://doi.org/10.1140/epjd/e2014-50146-y

Regular Article

## Hamiltonian structure of a drift-kinetic model and Hamiltonian
closures for its two-moment fluid reductions^{⋆}

Aix-Marseille Université, Université de Toulon, CNRS,
Centre de Physique Théorique, UMR 7332, 13288
Marseille,
France

^{a}
e-mail: tassi@cpt.univ-mrs.fr

Received:
25
February
2014

Received in final form:
28
April
2014

Published online:
24
July
2014

We address the problem of the existence of the Hamiltonian structure for an electrostatic
drift-kinetic model and for the related fluid models describing the evolution of the first
two moments of the distribution function with respect to the parallel velocity. The
drift-kinetic model, which accounts for background density and temperature gradients as
well as polarization effects, is shown to possess a noncanonical Hamiltonian structure.
The corresponding Poisson bracket is expressed in terms of the fluid moments and it is
found that the set of functionals of the zero order moment forms a sub-algebra, thus
automatically leading to a class of one-moment Hamiltonian fluid models. In particular, in
the limit of weak spatial variations of the background quantities, the
Charney-Hasegawa-Mima equation, with its Hamiltonian structure, is recovered. For the set
of functionals of the first two moments, which, unlike the case of the Vlasov equation,
turns out not to form a sub-algebra, we look for closures that lead to a closed Poisson
bracket restricted to this set of functionals. The constraint of the Jacobi identity turns
out to select the adiabatic equation of state for an ideal gas with one-degree-of-freedom
molecules, as the only admissible closure in this sense. When the so called
*δf* ordering
is applied to the model, on the other hand, a Poisson bracket is obtained if the second
order moment is a linear combination of the first two moments of the total distribution
function. By means of this procedure, three-dimensional Hamiltonian fluid models that
couple a generalized Charney-Hasegawa-Mima equation with an evolution equation for the
parallel velocity are derived. Among these, a model adopted by Meiss and Horton [Phys.
Fluids **26**, 990 (1983)] to describe drift waves coupled to ion-acoustic waves,
is obtained and its Hamiltonian structure is provided explicitly.

*© EDP Sciences, Società Italiana di Fisica, Springer-Verlag
2014*