Le système de Vlasov-Poisson effectif pour les plasmas fortement magnétisés

We study the finite Larmor radius regime for the Vlasov-Poisson system. The magnetic field is assumed to be uniform. We investigate this non-linear problem in the two-dimensional setting. We derive the limit model by appealing to gyro-average methods (cf. [1], [2]). We indicate the explicit expression of the effective advection field, entering the Vlasov equation, after substituting the self-consistent electric field, obtained by the resolution of the averaged (with respect to the cyclotronic time scale) Poisson equation. We emphasize the Hamiltonian structure of the limit model and present its properties: conservation of mass, of kinetic energy, of electric energy, etc.