A Dissipative Model of Plasma Equilibrium in Toroidal Systems

Abstract

In order to describe a steady-state plasma equilibrium in tokamaks, stellarators or other non-axisymmetric configurations, the model of ideal MHD with isotropic plasma pressure is widely used: 0 = - ∇p + j x B . The ideal MHD-model of a toroidal plasma equilibrium requires the existence of closed magnetic surfaces. Several numerical codes have been developed in the past to solve the three-dimensional equilibrium problem, but so far no existence theorem for a solution has been proved. Another difficulty is the formation of magnetic islands and field line ergodisation, which can only be described in terms of ideal MHD if the plasma pressure is constant in the ergodic region. In order to describe the formation of magnetic islands and ergodisation of surfaces properly, additional dissipative terms have to be incorporated to allow decoupling of the plasma and magnetic field. In a collisional plasma, viscosity and inelastic collisions introduce such dissipative processes. In the model used a friction term proportional to the velocity v of the plasma is included. Such a term originates from charge exchange interaction of the plasma with a neutral background. With these modifications, the equilibrium problem reduces to a set of quasilinear elliptic equations for the pressure, the electric potential and the magnetic field. The paper deals with an existence theorem based on the Fixed-Point method of Schauder. It can be shown that a self-consistent and unique equilibrium exists if the friction term is large and the plasma pressure is sufficiently low. The essential role of the dissipative terms is to remove the singularities of the ideal MHD model on rational magnetic surfaces. The problem has a strong similarity to Bénard cell convection, and consequently a similar behaviour such as bifurcation and exchange of stability is expected.