Expansion of a singularly perturbed equation with a two-scale converging convection term
Provides a theoretical framework for modeling high-amplitude convection in plasma physics, but the results are incremental and limited to linear cases.
The paper develops a model hierarchy for a singularly perturbed convection equation with two-scale converging convection terms, deriving an asymptotic expansion in ε and applying it to the linear Vlasov equation in magnetic confinement fusion and charged particle beam contexts.
In many physical contexts, evolution convection equations may present some very large amplitude convective terms. As an example, in the context of magnetic confinement fusion, the distribution function that describes the plasma satisfies the Vlasov equation in which some terms are of the same order as $ε^{-1}$, $ε\ll 1$ being the characteristic gyrokinetic period of the particles around the magnetic lines. In this paper, we aim to present a model hierarchy for modeling the distribution function for any value of $ε$ by using some two-scale convergence tools. Following Frénod \\& Sonnendrücker's recent work, we choose the framework of a singularly perturbed convection equation where the convective terms admit either a high amplitude part or a an oscillating part with high frequency $ε^{-1} \gg 1$. In this abstract framework, we derive an expansion with respect to the small parameter $ε$ and we recursively identify each term of this expansion. Finally, we apply this new model hierarchy to the context of a linear Vlasov equation in three physical contexts linked to the magnetic confinement fusion and the evolution of charged particle beams.