Multipole tokamak equilibria

Abstract

Solutions are given for axisymmetric MHD-equilibria obtained by superposing the magnetic field of the plasma currents and the field of current-carrying conductors situated outside the plasma. These equilibria were numerically calculated by several different iteration methods for solving the non-linear free boundary problem. They can describe experiments in which the discharge time is long compared with the diffusion time of the magnetic field through the external conducting walls. Equilibria were obtained both for approximately circular plasma cross-sections and strongly elongated (elliptical) ones. Such calculations are of interest for constructing axisymmetric tokamak divertors, as well as for producing elongated crosssections, which have been postulated to be stable for higher values of plasma pressure. The results presented here are restricted to the case of zero plasma pressure and a particular current distribution in order to reduce the number of free parameters and allow more detailed discussion of the dependence of the equilibrium configurations on the arrangement of the external conductors and the currents.

Numerical determination of axisymmetric toroidal magnetohydrodynamic equilibria

“…Hagenow and Lackner (see Feneberg and Lackner [16]), who devel oped codes similar to those described here. Solution techniques utilizing similar iterative methods have been reported by Cenacchi et al, [17] and Sakurai et al, [18].…”

Analytical Solution of the Three-Dimensional Laplace Equation in Terms of Linear Combinations of Hypergeometric Functions

“…As it is proved in [26] and subsequently reported by Crisanti [19], the Laplace Equation (10) in the cap-cyclide coordinates admits a quasi-separable solution of the type…”

“…The problem of finding the solution of the Grad-Shafranov equation was also addressed from the numerical point of view, through the development of some predictive equilibrium codes [10][11][12][13][14]. Semi-analytical methods have also been adopted in [15,16].…”

Analytical solution of the Grad Shafranov equation in an elliptical prolate geometry

“…The employment of a system of toroidal coordinates, which has the flux surfaces as the radial coordinate, has allowed us to develop robust codes with the mapping solution expanded on the momentum (Boozer 1980; Lao, Hirschmann & Wieland 1981; Brambilla 2003). Predictive equilibrium codes have been developed, through the numerical resolution (Feneberg & Lackner 1973; Cenacchi, Galvao & Taroni 1976; Blum, Le Foll & Thooris 1981; Albanese, Blum & De Barbieri 1987; Albanese, Ambrosino & Mattei 2015) and/or the adoption of a semi-analytical approach to deal with (Alladio & Crisanti 1986; Alladio et al. 1991) the Grad Shafranov equation, and have been then employed to design the planned plasma configuration of practically all ongoing tokamak experiments.…”