Section: New Results

Mathematical theory of reduced MHD models

Participant : Hervé Guillard.

One of the fundamental model used for fusion plasma simulations is the magnetohydrodynamic (MHD) model. However, in practice, many theoretical and numerical works in this field use specific approximations of this model known as reduced MHD models. These models assume that in the presence of a strong magnetic field, the main dynamic reduces to incompressible motion in the plane perpendicular to the dominating magnetic field and to the propagation of Alfvén waves in the magnetic field direction. In the framework of the slab approximation for large aspect ratio tokamaks ($R/a>>1$ where $R$ and $a$ are respectively the major and minor radius of the machine) we have studied last year the validity of this assumption using techniques coming from the asymptotic theory of hyperbolic equations with a large parameter. In particular, we have proved that the solutions of the full MHD system converge in a weak sense to the solutions of an appropriate reduced model even in the presence of ill-prepared initial data. This work continues with a tentative to relax the large aspect ratio assumption that is not verified in modern machines.