Hamiltonian canonical formulation of Hall-magnetohydrodynamics: Toward an application to weak turbulence theory

The different levels of description of fluid media [e.g., magnetohydrodynamics (MHD), Hall-magnetohydrodynamics, bi-fluid,...] are commonly known under the form of Newtonian systems of equations. Nevertheless, this form proves to be ill-suited to derive a fully analytical weak turbulence theory of these media, due to the well-known complexity of the calculations implied. For such studies, therefore, a more appropriate mathematical frame needs to be found and this is shown to be the Hamiltonian formalism, even though it can often appear difficult to handle. The goal of this paper is to look for Hamiltonian formulations for the different levels of the fluid description of a plasma using the variational principle. Starting from the bi-fluid system, it is shown that such a formulation can be obtained by combining the Lagrangians already used for describing: (i) the motion of a charged particle in an electromagnetic field; (ii) the evolution of an electromagnetic field in presence of sources; (iii) the motion of a neutral fluid (Clebsch variables). The equivalence of the obtained description in terms of the generalized-Clebsch variables to the familiar Newtonian formulation is discussed. It is shown that each solution of the Hamiltonian system is also a solution for the Newtonian one, but that the converse is not true. The origin and the implication of this restriction are discussed. Reducing the Hamiltonian formulation obtained for the bi-fluid system to lower orders of the fluid approximations is then shown to be mandatory when one tries to obtain analytical results for linear waves and nonlinear wave-wave couplings. It is shown that this goal can be reached in two steps. The first one leads to a "reduced bi-fluid" system, which is identical to the bi-fluid one when the displacement current is neglected but the electron inertia is still working. The number of linear modes then goes down from six to three. The second step, leading to the Hall-MHD system, consists in neglecting the electron mass. It is demonstrated that the only four generalized Clebsch variables are sufficient to describe the full Hall-MHD dynamics. Some future applications of such a powerful formalism are outlined. (C) 2003 American Institute of Physics.