Applied Mathematics and Mathematical Physics Seminar: Calan Atanasiu

Abstract
Magneto-hydrodynamic equilibrium and stability calculations (analytical and numerical) for real diverted tokamak configurations are presented. (a) Two families of exact analytical solutions of the Grad-Shafranov equation are presented by specifying the highest polynomial dependence of the plasma current density on the solution in such a way that the Grad-Shafranov equation becomes a linear inhomogeneous differential equation. (b) By introducing a "cast function" in a classical flux coordinate system, in the presence of a separatrix, the solution of the equilibrium equation - the unknown moments - is determined by the difference between the real flux surface contours and those described by the cast functions only. Thus, the necessary number of moments is small enough to make computations time-efficient. (c) For instability calculation of tearing and external kink type, the expression of the potential energy has been written in terms of the perturbation of the flux function, and performing an Euler minimization, a system of ordinary differential equations in that perturbation has been obtained. For a diverted configuration, the usual vanishing boundary conditions for the perturbed flux function at the magnetic axis and at infinity can no longer be used. An approach to fix "natural" boundary conditions for the perturbed flux function just at the plasma boundary has been developed; this replaces the vanishing boundary conditions at infinity. Special attention is given to the stabilization of external kink modes in the presence of a conducting wall - the resistive wall modes, the most dangerous instability for the future tokamak reactors.