Imposing quasineutrality on electrostatic plasmas via the Dirac theory of constraints

Abstract

We present a method for imposing quasineutrality and, more generally, charge density conservation in the Vlasov-Poisson (VP) and Vlasov-Ampère (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.

Figures (8)

• Figure 1: Initial conditions of the distribution functions $f_e$ and $f_i$ in phase space. The electron distribution forms two separated beams since $V_e$ is large enough in \ref{['f_e']}, whereas the two ion beams are so close to each other effectively forming a single beam with zero macroscopic velocity. This broad, centralized ion beam was found to favor numerical stability over longer simulation times.
• Figure 2: Snapshots of the electron and ion distribution functions in phase space $(x,v)$ at different times. The left column corresponds to the VP case, and the right column to the QN case. A distinct evolution of $f_e$ is observed between the two scenarios, especially during the nonlinear phase.
• Figure 3: Left: Evolution of the average charge density modulus $\langle|\rho|\rangle$ in the standard VP scenario (solid red line) vs the Dirac-constrained QN scenario (dashed blue line). The QN charge density, although non-zero, remains consistently three orders of magnitude smaller than in the VP case, even in the vortex saturation stage (right).
• Figure 4: Left: Evolution of $\langle|\partial_x J|\rangle$ in the standard VP scenario (solid red line) vs the Dirac-constrained QN scenario (dashed blue line). Although non-zero, this quantity remains consistently at least three orders of magnitude smaller than in the VP case, even in the vortex saturation stage (right).
• Figure 5: Relative energy error (left) and relative particle error (right) in the Vlasov-Poisson simulation. The particle number and energy are not conserved with high precision due to the non-conservative nature of the simulation algorithm, interpolation errors, and the applied filtering method.