Navier-Stokes hierarchies of reduced MHD models in Tokamak geometry

We study the closure of reduced MHD models. We show how to modify the entropy moment methods to obtain a hierarchy of models of Navier-Stokes type with a correct energy balance. Our procedure is based on the entropy and well adapted to the complicated geometry of the torus. A new correction term due to the geometry is derived, what is important for the stability of the model. We prove the existence of a weak solution to some of these models by similarity with Navier-Stokes theory in potential formulation. We finally obtain a comparison principle between all these models.