Hamiltonian fluid closures of the Vlasov-Amp{\`e}re equations: from water-bags to N moment models

TL;DR

This paper develops Hamiltonian fluid closures for the Vlasov-Ampère equations using water-bag models, establishing a link to N-moment fluid models with preserved Hamiltonian structure and Casimir invariants.

Contribution

It introduces a systematic method to derive Hamiltonian fluid models from water-bag representations, extending to arbitrary numbers of fields and moments.

Findings

Hamiltonian structures for 1, 2, and 3 water-bag models are provided.

A general procedure for N-field Hamiltonian fluid models is proposed.

The thermodynamic interpretation of these models is discussed.

Abstract

Moment closures of the Vlasov-Amp{\`e}re system, whereby higher moments are represented as functions of lower moments with the constraint that the resulting fluid system remains Hamiltonian, are investigated by using water-bag theory. The link between the water-bag formalism and fluid models that involve density, fluid velocity, pressure and higher moments is established by introducing suitable thermodynamic variables. The cases of one, two and three water-bags are treated and their Hamiltonian structures are provided. In each case, we give the associated fluid closures and we discuss their Casimir invariants. We show how the method can be extended to an arbitrary number of fields, i.e., an arbitrary number of water-bags and associated moments. The thermodynamic interpretation of the resulting models is discussed. Finally, a general procedure to derive Hamiltonian N-field fluid models is proposed.