Determination of stable three-dimensional equilibria

A procedure has been devised and is discussed for the determination of stable three-dimensional equilibria. It is shown that, in contrast to the simple second-order Grad-Shafranov equation for two-dimensional equilibria, the magnetostatic equations define a quasi-linear fourth-order system in three dimensions. Using a variational approach, an iterative method is devised for determining such equilibria. The method defines a displacement field, dependent on the residual force, with which the pressure and magnetic fields are moved. Convergence is optimized using the Chebyshev semi-iterative method. Solutions converge towards a stable equilibrium. Given an unstable equilibrium, the fastest growing instability will evolve where the linear growth rate can be evaluated. These solutions compare well with conventional normal-mode analysis. The further non-linear evolution of such an instability leads either to the gross destruction of the plasma column or to the formation of secondary stable equilibria. In the tokamak regime, the solutions illustrate the gross destruction of the plasma if external kink modes are permitted. With the safety factor q exceeding unity externally, any internal modes which can grow according to linear theory are found to deform the internal surfaces grossly, while the outer surfaces are relatively unperturbed. The existence of secondary stable bubble equilibria is found. An extreme example of this is illustrated where a plasma column develops non-linearly into a stable shell. (author)