MHD model

The author's goal is to provide a physical understanding of the ideal MHD model which includes: (1) a basic description of the model, (2) a derivation starting from a more fundamental kinetic model, and (3) a discussion of its range of validity. The ideal MHD model is a single-fluid model that describes the effects of magnetic geometry on the macroscopic equilibrium and stability properties of fusion plasmas. The model is derived in a straight forward manner by forming the mass, momentum, and energy moments of the Boltzmann equation. The moment equations reduce to ideal MHD with the introduction of three critical assumptions: high collisionality, small ion gyro radius, and small resistivity. An analysis of the validity conditions shows that the collision-dominated assumption is never satisfied in plasmas of fusion interest. The remaining two conditions are satisfied by a wide margin. A careful examination of the collision-dominated assumption shows that those particular parts of ideal MHD treated inaccurately (i.e., the parallel momentum and energy equations), play little, if any practical role in MHD equilibrium and stability. These equations primarily describe compression and expansion of a plasma whereas most MHD instabilities involve incompressible motions. The model is incorrect only where it does not matter. This realization leads to the introduction of a modified MHD model known as collisionless MHD which makes predictions nearly identical to collision-dominated assumption. It is thus valid for plasmas of fusion interest. The derivation follows from an analysis of single-particle guiding center motion in a collisionless plasma and the subsequent closure of the system by the heuristic assumption that the motions of interest are incompressible