Imposing quasineutrality on electrostatic plasmas via the Dirac theory of constraints

TL;DR

This paper introduces a Hamiltonian-based method using Dirac constraints to enforce quasineutrality in electrostatic plasma models, ensuring charge conservation and modifying plasma dynamics.

Contribution

It develops a systematic approach to impose quasineutrality via Dirac brackets, eliminating electric fields and adding force terms in the Vlasov equations.

Findings

Charge density conservation is verified numerically.

Quasineutrality enforcement alters plasma dynamics.

The method identifies forces needed for quasineutrality.

Abstract

We present a method for imposing quasineutrality and, more generally, charge density conservation in the Vlasov-Poisson (VP) and Vlasov-Amp\`ere (VA) systems, which describe electrostatic plasma dynamics, by applying the Dirac theory of constraints. Leveraging the Hamiltonian field formulations of the VP and VA models, we construct generalized Dirac brackets using the Dirac algorithm. The resulting constrained systems enforce charge density conservation, and consequently quasineutrality, given that the initial charge density is zero, through new advection terms in the Vlasov equations involving generalized-force terms, while the electric field is eliminated from the constrained Vlasov dynamics. To verify charge density conservation we conduct one-dimensional numerical experiments using a semi-Lagrangian method, demonstrating that the enforcement of the quasineutrality constraint significantly modifies the dynamics. This approach enables us to identify the forces required to enforce quasineutrality, offering a systematic way to assess the validity of the quasineutral approximation across different kinetic scales.