Two-scale expansion of a singularly perturbed convection equation

In magnetic fusion, a plasma is constrained by a very large magnetic field, which introduces a new time scale. namely the period of rotation of the pa rticles around the magnetic field lines. This new time scale is very restri ctive for numerical simulation. which makes it important to find approximat e models of the Vlasov-Poisson equation where it is removed. The gyrokineti c models aim at exactly this. Such models have been derived in the physics literature for several decades now, but only in the last few years there ha ve been rigorous mathematical derivations. Those have only addressed the li mit when the magnetic field becomes infinite, We consider here the Vlasov e quation in different physical regimes for which small parameters are identi fied, and cast the obtained dimensionless equations into the abstract frame work of a singularly perturbed convection equation. In this framework we de rive an asymptotic expansion with respect to the small parameter of its sol ution, and characterize the terms of the expansion. The proofs make use of Allaire's two-scale convergence. (C) 2001 Editions scientifiques et medical es Elsevier SAS.